Download - e GMAT Number Properties
Unauthorized copying prohibited
Number Properties IQUANT LIVE SESSION
Session Layout
ConceptConcept
Test 3 GMAT 700+ QEven-Odd
1
ConceptConcept
Test 5 GMAT 700+ QPrimes2
ConceptConcept
Test 2 GMAT 700+ QLCM-GCD
3
Part 1EVEN-ODD NUMBERS
Unauthorized copying prohibited
Even Odd Numbers - Properties
Odd * Odd = Odd
Every EVEN number can be represented as 2n, where n is an integer
Every ODD number can be represented as 2n+1, where n is an integer
Basic Properties:
(Even)2 = (2n)2 = 4n2
Derived Properties:
Divisible by 4
Properties2 = 2 × 1 4 = 2 × 2
6 = 2 × 3 8 = 2 × 4
3 = 2 × 1 + 1 5 = 2 × 2 + 1
7 = 2 × 3 + 1 9 = 2 × 4 + 1
Even +/- Even = Even2 + 4 = 68 – 2 = 6
Even +/- Odd = Odd2 + 3 = 52 - 3 = -1
Odd +/- Odd = Even1 + 3 = 41 – 3 = -2
Even * Even = Even
Even * Odd = Even
2 × 4 = 8
2 × 3 = 6
3 × 5 = 15
(Even)n +/- (Even)n = Even +/- Even = Even
(Odd)n +/- (Odd)n = Odd +/- Odd = Even
(Even)n +/- (Odd)n = Even +/- Odd = Odd
Unauthorized copying prohibited
Test your Understanding
1. A2 + B2 = (Even, Odd, Cannot determine)
Answer: Consecutive => One Even, the other odd A2 + B2 = Even2 + Odd2
Odd
A, B, C, D are consecutive integers > 1. Then
2. A2 + B2 + C2= (Even, Odd, Cannot determine)
Answer: Consecutive => (Even, Odd, Even) OR (Odd, Even, Odd) A2 + B2 + C2 = (Even2 + Odd2 + Even2) OR (Odd2 + Even2 + Odd2) A2 + B2 + C2 = (Even + Odd + Even) OR (Odd + Even + Odd) A2 + B2 + C2 = Odd OR Even
Cannot Determine
Question = MCQ question, Answer choices: Even, Odd, Cannot be determined
Even Odd+
3. A2 + B3 + C3= (Even, Odd, Cannot determine) Answer: Cannot determine
Unauthorized copying prohibited
Test your Understanding
4. B7 - D7 = (Even, Odd, Cannot determine)
Answer: Consecutive => (Even, Odd, Even, Odd)
OR(Odd, Even, Odd, Even)
B7 - D7 = (Odd2 - Odd2) OR (Even2 - Even2) B7 - D7 = (Odd - Odd) OR (Even - Even) B7 - D7 = Even
Even
A, B, C, D are consecutive integers > 1. Then
Question = MCQ question, Answer choices: Even, Odd, Cannot be determined
5. A2 + B2 + C2 + D2 = (Even, Odd, Cannot determine) Answer: Even 2 Odd numbers + 2 Even numbers = Even
Unauthorized copying prohibited
Apply in GMAT Context: Question 1
Is an – bn odd, if a, b, and n are positive integers?
1. a and b are squares of consecutive natural numbers
2. a2 + b2 is odd
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Data Sufficiency Question Solving Process
Step 1: Understand the question
Step 2: Draw Inferences
Step 3: Analyze Statement 1 independently
Step 4: Analyze Statement 2 independently
Step 5: Analyze both Statements together
Is an – bn odd, if a, b, and n are positive integers?
1. a and b are squares of consecutive natural numbers
2. a2 + b2 is odd
Unauthorized copying prohibited
Question 1 – Steps 1 and 2 – Understand question statement and Draw Inferences
Question Statement - Is an – bn odd, if a and b are positive integers?
Given –
1. a, b, n > 0
2. a, b, n are integers
To find –
Is an – bn odd?
One of them is odd and the other is even
an – bn odd Even Odd
an
x
bn
x
a
b
Is an – bn odd? Is only one of a or b odd?
Power does not change the even/odd nature of a number• (Even)n = Even• (Odd)n = Odd
Unauthorized copying prohibited
Question 1 – Step 3 – Analyze Statement 1 Independently
Statement 1 - a and b are squares of consecutive natural numbers
Is an – bn odd?
Is only one of a or b odd?
Consecutive natural numbers P, P+1
P2, (P+1)2{a, b}
If P is even P+1 is odd
Each case is an even-odd pair
Answers the question – YES! Only one of a or b is odd
Statement 1 is sufficient
a, b integers >0
P2 is even (P+1)2 is odd
If P is odd P+1 is even
P2 is odd (P+1)2 is even
Unauthorized copying prohibited
Question 1 – Step 4 – Analyze Statement 2 Independently
Statement 2 - a2 + b2 is odd
(a,b) is an even-odd pair
Answers the question – YES! Only one of a or b is odd
Statement 2 is sufficient
Correct answer = Choice D = Either statement is sufficient to answer the question
Is an – bn odd?
Is only one of a or b odd?
a, b integers >0
a a2 b b2 a2 + b2
Even Even Even Even Even
Even Even Odd Odd Odd
Odd Odd Even Even Odd
Odd Odd Odd Odd Even
Unauthorized copying prohibited
Apply in GMAT Context: Question 2
If P = k3 – k, where k and P are positive integers, is P divisible by 4?
1. k = (10x)n + 54 where x and n are positive integers
2. (2n+1)k leaves a remainder when divided by 2; n is a positive integer
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Unauthorized copying prohibited
Question 2 – Steps 1 and 2 – Understand question statement and Draw Inferences
Question Statement - If P = k3 – k, where k and P are positive integers, is P divisible by 4?
?Given –
P = k3 – kTo find –
If P is divisible by 4.
= k(k2-1)
= k(k-1)(k+1)
= (k-1) k(k+1) = Product of 3 consecutive integersCase 1
Case 2
(k-1) : even
(k+1) : even
=2m
=2m + 2product has 4 P always divisible by 4
(k-1) : odd
(k+1) : odd
=2m
product has 2
P divisible by 4 if
k is divisible by 4
k: odd
k: evenP may or may not be divisible by 4
P is divisible by 4 if either1. k is odd or2. k is even & k is divisible by 4
Unauthorized copying prohibited
Question 2 – Step 3 – Analyze Statement 1 Independently
Statement 1: k = (10x)n + 54 where x and n are positive integers
Answers the question – P is divisible by 4
Statement 1 is sufficient
P = (k-1) k(k+1) Is P is divisible by 4?P is divisible by 4 if either1. k is odd or2. k is even & k is divisible by 4
k = (10x)n + 54
= 2n (5x)n + 54
even odd+
k is odd
Unauthorized copying prohibited
Question 2 – Step 4 – Analyze Statement 2 Independently
Statement 2: (2n+1)k leaves a remainder when divided by 2
Answers the question – P is divisible by 4
Statement 2 is sufficient
P = (k-1) k(k+1) Is P is divisible by 4?P is divisible by 4 if either1. k is odd or2. k is even & k is divisible by 4
(2n + 1) k is odd
oddodd
k is odd
Correct answer = Choice D = Either statement is sufficient to answer the question
Unauthorized copying prohibited
Apply in GMAT Context: Question 3
If P = k3 – k2, where k and P are positive integers, is P divisible by 4?
1. k = (10x)n + 54 where x and n are positive integers and n > 1.
2. (2n+1)k leaves a remainder when divided by 2; n is a positive integer
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Unauthorized copying prohibited
Question 2 vs. 3
If P = k3 – k, where k and P are positive integers, is P divisible by 4?
1. k = (10x)n + 54 where x and n are positive integers
2. (2n+1)k leaves a remainder when divided by 2; n is a positive integer
If P = k3 – k2, where k and P are positive integers, is P divisible by 4?
1. k = (10x)n + 54 where x and n are positive integers and n > 1.
2. (2n+1)k leaves a remainder when divided by 2; n is a positive integer
Q3
Q4
Unauthorized copying prohibited
Question 3 – Steps 1 and 2 – Understand question statement and Draw Inferences
Question Statement - If P = k3 – k2, where k and P are positive integers, is P divisible by 4?
?Given –
P = k3 – k2
To find –
If P is divisible by 4.
= k2(k-1)
Case 1
Case 2
(k-1) : even =2n
product has 4 P always divisible by 4(k-1) : odd
=2n
product has 2
P divisible by 4 if
(k-1) is divisible by 4
k: odd
k: even
P may or may not be divisible by 4
P is divisible by 4 if either1. k is even or2. k is odd & k-1 is divisible by 4
k2: odd
k2: even =(2n)2
Unauthorized copying prohibited
Question 3 – Step 3 – Analyze Statement 1 Independently
Statement 1: k = (10x)n + 54 where x and n are positive integers and n>1
Answers the question – P is divisible by 4
Statement 1 is sufficient
P = k2(k-1) Is P is divisible by 4?
k = (10x)n + 54
= 2n (5x)n + 54
even odd+
k is odd
P is divisible by 4 if either1. k is even or2. k is odd & k-1 is divisible by 4
Is k – 1 divisible by 4?
k – 1 = (10x)n + 54 - 1
n > 1 n ≥ 2 100xn 625 1+ -=
Tens and units digit = 24
(k-1) is divisible by 4
Unauthorized copying prohibited
Question 3 – Step 4 – Analyze Statement 2 Independently
Statement 2: (2n+1)k leaves a remainder when divided by 2; n is a positive integer.
Does not answer the question – P is divisible by 4
Statement 2 is NOT sufficient
(2n + 1) k is odd
oddodd
k is odd
Correct answer = Choice A = Statement 1 is sufficient but statement 2 is not
Is P is divisible by 4?
P is divisible by 4 if either1. k is even or2. k is odd & k-1 is divisible by 4
Don’t know if k-1 is divisible by 4
P = k2(k-1)
Part 2PRIMES
Unauthorized copying prohibited
Prime Number & Factors
1. A prime number is a positive integer that has
exactly two different positive factors, 1 and
itself. Examples: 2, 3, 5, 7 . . .
2. 0 and 1
are neither Prime nor Composite.
--- Do not have TWO different positive factors
3. Every positive integer K can be expressed as K
= P1m × P2
n × P3r ……, where P1, P2, P3 …… are
prime factors and m, n , r are non-negative
integers
Basic Properties2 is the smallest Prime number
2 and 3 are the only consecutive numbers
that are prime
2 is the only even Prime number
4 = 22
6 = 2× 3
8 = 23
10 = 2 × 5
12 = 22 × 3
18= 2 × 32
1000 = 23 × 53
2400 = 25 × 3 × 52
Prime Factorization
V/S
Consecutive prime numbers
Unauthorized copying prohibited
Prime Number & Factors
Total number of factors = (m+1) (n +1)(r+1) . . .
K = P1m × P2
n × P3r . . . where P1, P2, P3 …… are prime factors and m, n , r
are non-negative integers
Kn will have the same prime factors as K
PnKm will have the same prime factors as K,
if P is a prime factor of K
Prime Factors
Total Factors
Possible powers of P1 in a factor: P10, P1
1, P12 . . . P1
m
Possible powers of P2 in a factor: P20, P2
1, P22 . . . P2
n
Possible powers of P3 in a factor: P30, P3
1, P32 . . . P3
r
(m+1) values
(n+1) values
(r+1) values
Eg: 62 & 6 have the same prime factors: 2 and 3
Eg: 23 * 62 have the same prime factors as 6: 2 and 3
Unauthorized copying prohibited
Prime Number & Factors – Test your Understanding
Total number of factors = (m+1) (n +1)(r+1) . . .
K = P1m × P2
n × P3r . . . where P1, P2, P3 …… are prime factors and m, n , r
are non-negative integers
Q: How many prime factors does K have if the total number of factors of K is:
a. 1
b. 2
c. 3
d. 4
e. 5
f. 6
g. 7
K = 1
K = P1
K = P12
K = P1 × P2
K = P14
K = P12 × P2
K = P16
0
1
1
2
1
2
1
A perfect square will
have an odd number
of factors
Unauthorized copying prohibited
Factors/Prime Factors- Test your Understanding
If X has 3 prime factors and 8 total factors, then how many prime factors will Xn have? (FIB)
Question 1
If X has 3 prime factors and 8 total factors, then how many factors will Xn
have? (FIB)
Question 3
If K is a factor of positive integer X that has 3 Prime factors and 8 total factors, then how Prime factors does K2 Xn
have?
Question 2
Answer– 3
Answer – 3
Answer – (n+1)3
Kn will have the same
prime factors as K
X = P1 × P2 × P3
X = P1 × P2 × P3
Xn = P1n × P2
n × P3n
If K is a factor of positive integer X that has 8 total factors, then how Prime factors does K2 Xn have?
Question 4
X = P1 × P2 × P3 X = P17 X = P1
3 × P2 or or
Answer – Cannot Determine
Unauthorized copying prohibited
Apply in GMAT Context: Question 1
Is an – bn odd, if a, b, and n are positive integers?
1. a and b are squares of consecutive prime numbers
2. a2 + b2 is odd
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
1. a and b are squares of consecutive natural numbers
2. a2 + b2 is odd
Even- Odd Question 1
Unauthorized copying prohibited
From Even- Odd Question 1
Question 1 – Step 3 – Analyze Statement 1 Independently
Statement 1 - a and b are squares of consecutive prime numbers
Is an – bn odd? Is only one of a or b odd?
Consecutive natural numbers P, P+1
{a, b} {P2, (P+1)2}
Each case is an even-odd pair
Answers the question – YES! Only one of a or b is odd
Statement 1 is sufficient
Consecutive prime numbers 2, 3, 5, 7…
Squares 4, 9, 25, 49…
{a, b} {4, 9} {9, 25} {25, 49}odd-oddeven-odd
Two possible scenarios as shown
Cannot answer the question – is only one of a or b odd?
Statement 1 is NOT sufficient
Understanding Information Given in the question is very critical.
Choice B Choice D
If P is even P+1 is odd
P2 is even (P+1)2 is odd
If P is odd P+1 is even
P2 is odd (P+1)2 is even
Unauthorized copying prohibited
Apply in GMAT Context: Question 2
Is an – bn + cn + dn odd, if a, b, c, and d are positive integers >1?
1. a, b, c, and d are squares of consecutive prime numbers
2. a4 when divided by 200 has the quotient 1
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Unauthorized copying prohibited
Question 2 – Steps 1 and 2 – Understand question statement and Draw Inferences
Question Statement - Is an – bn + cn + dn odd, if a, b, c, and d are positive integers >1?
Given –
1. a, b, c, and d > 1
2. a, b, c, and d are integers
To find –
Is an – bn + cn + dn odd?
Is 1 or are 3 of the 4 numbers odd?
Power does not change the even/odd nature of a number• (Even)n = Even• (Odd)n = Odd
• a ± b ± c ± d = e or o?
• o ± o ± o ± o = e
• e ± o ± o ± o = o
• e ± e ± e ± o = o
• e ± e ± e ± e = e
• e ± e ± o ± o = e
An expression with sum or difference of integers is odd if odd number of terms are odd.
Unauthorized copying prohibited
Question 2 – Step 3 – Analyze Statement 1 Independently
Statement 1 - a, b, c, and d are squares of consecutive prime numbers
Consecutive prime numbers 2, 3, 5, 7, 11, 13…
Squares 4, 9, 25, 49, 121, 169…
{a, b, c, d} {4, 9, 25, 49}
e, o, o, o
Does not answer the question – the expression can be either even or odd
Statement 1 is not sufficient
a, b, c, and d > 1Is an – bn + cn + dn odd?
Is 1 or are 3 of the 4 numbers odd?
{9, 25, 49, 121} {25, 49, 121, 169}
o, o, o, o o, o, o, o
Unauthorized copying prohibited
Question 2 – Step 4 – Analyze Statement 1 Independently
Statement 2 – a4 when divided by 200 has the quotient 1
Does not answer the question – the expression can be either even or odd
Statement 2 is not sufficient
a, b, c, and d > 1Is an – bn + cn + dn odd?
Is 1 or are 3 of the 4 numbers odd?
a4/200 has quotient 1
200 < a4 < 400
a = 2
a = 3
a4 = 16
a4 = 81
a = 4 But we don’t know if b, c, and d are odd or even
a ≠ 1
a = 4 a4 = 256
a = 5 a4 = 625
Unauthorized copying prohibited
Question 2 – Step 5 – Analyze Statements 1 & 2 Together
Statements 1& 2
Answers the question
Both together are sufficient
a, b, c, and d > 1Is an – bn + cn + dn odd?
Is 1 or are 3 of the 4 numbers odd?
a = 4
Statement 1 {a, b, c, d} {4, 9, 25, 49}
e, o, o, o
{9, 25, 49, 121} {25, 49, 121, 169}
o, o, o, o o, o, o, o
Statement 2
{a, b, c, d} = {4, 9, 25, 49}
e, o, o, o
3 of the numbers are odd
Correct answer = Choice C = Both together are sufficient
Unauthorized copying prohibited
Apply in GMAT Context: Question 3
What is the remainder when b is divided by a, if a and b are consecutive perfect squares and b is greater than a?
1. Both a and b are squares of prime numbers.
2. Both a and b have 3 positive factors.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Unauthorized copying prohibited
Question 3 – Steps 1 and 2 – Understand question statement and Draw Inferences
Question Statement - What is the remainder when b is divided by a, if a and b are consecutive perfect squares and b is greater than a?
?Given –
1. b > a
2. a and b are consecutive perfect squares
To find –
Remainder of b/a = ?
Need to know values of b and a
Consecutive numbers 1, 2, 3, 4, 5…
Squares 1, 4, 9, 16, 25…
{a, b} {1, 4} {4, 9} {9, 16} {16, 25} …
Unauthorized copying prohibited
Question 3– Step 3 – Analyze Statement 1 Independently
Statement 1 - Both a and b are squares of prime numbers.
Remainder of b/a = ?
Need to know values of a and b
Values of a, b known
Answers the question – remainder can be calculated
Statement 1 is sufficient
Per question {a, b} squares of consecutive numbers
Prime numbers 2, 3, 5, 7…
Squares of prime numbers that are consecutive
{a, b} = {1, 4} {4, 9} {9, 16} {16, 25}…
a, b are consecutive squares
{2,3}
Unauthorized copying prohibited
Question 3 – Step 4 – Analyze Statement 2 Independently
Statement 2 - Both a and b have 3 positive factors.
Values of a, b known
Answers the question – remainder can be calculated
Statement 2 is sufficient
a = P12m × P2
2n × …
{a, b} = {4, 9}
Remainder of b/a = ?
Need to know values of a and b{a, b} = {1, 4} {4, 9} {9, 16} {16, 25}…
b = P’12r + P’2
2s +…
a, b are consecutive squares
Total number of factors of a = (2m+1)(2n+1) × …
= (2m+1)(2n+1) × …3
= (2m+1)(2n+1) × …(2x1+1)(2x0+1)
a = P12
a is square of a prime number
Similarly b is square of a prime number
{2, 3} are the only consecutive numbers that are prime
Correct answer = Choice D = Either statement is sufficient to answer the question
Square of prime number has 3 factors
m = 1; n . . . = 0
Unauthorized copying prohibited
Apply in GMAT Context: Question 4
How many distinct prime factors does √Q have, if Q is a perfect square of a positive integer?1. Q is odd and 8Q8 has four distinct prime factors2. 8Q and Q2 do not have the same set of prime factors
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Unauthorized copying prohibited
Question 4 – Steps 1 and 2 – Understand question statement and Draw Inferences
Question Statement - How many distinct prime factors does √Q have, if Q is a perfect square of a positive integer?
? Given –
Q is a perfect square of a positive integer
To find –
Number of prime factors of √Q
Number of Prime Factors of Q
NOT Total number of factors!
√Q is a positive integer
Q = (√Q)2
Kn will have the same prime factors as K
(√Q)2 will have the same prime factors as √Q
Unauthorized copying prohibited
Question 4 – Step 3 – Analyze Statement 1 Independently
Statement 1: Q is odd and 8Q8 has four distinct prime factors
Answers the question – Q has 3 distinct prime factors
Statement 1 is sufficient
√Q is a positive integer # of Prime factors of √Q # of Prime factors of Q
8Q8 = 23Q8
Prime factors of 8Q8 = 2, Prime factors of Q8
Q is odd
Prime factors of Q
Odd
4 1
32
If Q were even
2 is a prime factor of Q
Q has 4 prime factors
Kn will have the same prime factors as K
Unauthorized copying prohibited
Question 4 – Step 4 – Analyze Statement 2 Independently
Statement 2: 8Q and Q2 do not have the same set of prime factors
Statement 2 is NOT sufficient
√Q is a positive integer # of Prime factors of √Q # of Prime factors of Q
8Q = 23Q
Prime factors of 8Q = 2, Prime factors of Q
Prime factors of Q2
Xn has the same prime factors as X
Q is oddOdd
How Many?
Correct answer = Choice A = Statement 1 is sufficient but statement 2 is not
Unauthorized copying prohibited
IF STATEMENT II IS NOT DECOUPLED FROM STATEMENT 1
Question 4 – ERROR ALERT!!!
Prime factors of 8Q8 = 2, Prime factors of Q8
Prime factors of Q
Odd
4 1
32
8Q = 23Q
Prime factors of 8Q = 2, Prime factors of Q
Prime factors of Q2
Q is oddOdd
How Many?
Statement 1 Statement 2
You may assume that there are 4 factors of 8Q
And you may consider Statement 2 to be sufficient as well
Answer = Choice D = Either statement is sufficient to answer the question
Unauthorized copying prohibited
Apply in GMAT Context: Question 5
How many distinct factors does √Q have, if Q is a perfect square of a positive integer?1. Q is odd and 8Q8 has four distinct prime factors2. 8Q and Q2 do not have the same set of prime factors
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Unauthorized copying prohibited
Question 5 – Steps 1 and 2 – Understand question statement and Draw Inferences
Question Statement - How many distinct factors does √Q have, if Q is a perfect square of a positive integer?
Given –
Q is a perfect square of a positive integerTo find –
Number of distinct factors of √Q
Each prime factor of Q has even power
Q = P12m × P2
2n × P32q × . . .
Where P1 , P2 , P3 . . . are prime numbers
m, n, q . . . are positive integers
Total number of factors of Q = (2m+1)(2n+1)(2q+1) …
√Q = P1m × P2
n × P3q × . . .
(m+1)(n+1)(q+1) …
Unauthorized copying prohibited
Question 5 – Step 3 – Analyze Statement 1 Independently
Statement 1: Q is odd and 8Q8 has four distinct prime factors
Statement 1 is NOT sufficient
Q = P12m × P2
2n × P32q × . . . # of factors of √Q (m+1)(n+1)(q+1) . . .
8Q8 = 23Q8
Prime factors of 8Q8 = 2, Prime factors of Q8
Q is odd
Prime factors of Q
Xn has the same prime factors as X
Odd
4 1
32
Q = P12m × P2
2n × P32q m, n, q
Unauthorized copying prohibited
Question 5 – Step 4 – Analyze Statement 2 Independently
Statement 2: 8Q and Q2 do not have the same set of prime factors
Statement 2 is NOT sufficient
8Q = 23Q
Prime factors of 8Q = 2, Prime factors of Q
Prime factors of Q2
Xn has the same prime factors as X
Q is oddOdd
How Many?
Q = P12m × P2
2n × P32q × . . . # of factors of √Q (m+1)(n+1)(q+1) . . .
What powers?
Unauthorized copying prohibited
Question 5 – Step 5 – Analyze Statements 1 & 2 Together
Statements 1& 2
Statement 1
Statement 2
Correct answer = Choice E = Both together are not sufficient
Q = P12m × P2
2n × P32q × . . . # of factors of √Q (m+1)(n+1)(q+1) . . .
Q = P12m × P2
2n × P32q
P1 ≠ P2 ≠ P3 ≠ 2
Q is oddStatement 1 + 2 are NOT sufficient
m, n, q
3 prime factors, all odd
Part 3LCM AND GCD
Unauthorized copying prohibited
Concept Slide on GCD and LCM
GREATEST COMMON DENOMINATOR LEAST COMMON MULTIPLE
40
1. Find the prime factors of the given numbers
40 = 2 × 2 × 2 × 5 98 = 2 × 7 × 7
2. Write the prime factors in exponential form
98 = 2 × 7240 = 23 × 5
3. Pick the SMALLEST power of each prime factor
4. Multiply the numbers from 3.
GCD
2 5 71 0 0
21 × 50 × 70 = 2
1. Find the prime factors of the given numbers
40 = 2 × 2 × 2 × 5 98 = 2 × 7 × 7
2. Write the prime factors in exponential form
98 = 2 × 7240 = 23 × 5
3. Pick the GREATEST power of each prime factor
4. Multiply the numbers from 3.
LCM
2 5 73 1 2
23 × 51 × 72
98× = GCD × LCM
Unauthorized copying prohibited
GCD - Test your Understanding
If GCD of two numbers (both integers, greater than 1) is 1, then which of the following can be true?
1. They are prime.2. They are consecutive.3. They do not have a common prime factor4. They do not have a common factor other than 1
I. Only 1II. Only 2III. Only 3 and 4IV. Only 2 and 3V. 1, 2, 3 and 4
Answer V – 1,2,3,4
If GCD of two numbers (both integers, greater than 1) is 1, then which of the following must be true?
1. They are prime.2. They are consecutive.3. They do not have a common prime factor4. They do not have a common factor other than 1
I. Only 1II. Only 2III. Only 3 and 4IV. Only 1 and 4V. 1, 2, 3 and 4
Answer III – Only 3 and 4
Question 1 Question 2
Question = MCQ question, Answer choices: A, B, C, D, E
Unauthorized copying prohibited
LCM - Test your Understanding
Answer: Only #3 and #5 Answer: 1, 2, 4, 5
If the LCM of two integers a, b (where b> a and a>1) is b, then which of the following can be true?
1. Both a and b can be Prime Numbers.2. Both a and b can be consecutive integers.3. All prime factors of a must be prime
factors of b.4. All prime factors of b must be prime
factors of a.5. b must be a multiple of a.
Question 1 (MAQ)
If the LCM of two integers a, b (where b> a and a>1) is a*b, then which of the following can be true?
1. Both a and b can be Prime Numbers.2. Both a and b can be consecutive integers.3. All Prime factors of a must be Prime
factors of b.4. a and b do not share any Prime factors.5. a and b do not have a common factor
Question 2 (MAQ)
Unauthorized copying prohibited
LCM- Test your Understanding
How prime factors does K2 Xn have, if K is Prime, X has 3 Prime factors and the LCM of K and X is KX?
Question 4
Answer = 3+ 1 = 4
Question 3
If the LCM of two integers a, b where b> a and a>1 is a*b/5, then what is the GCD of a & b?
Answer: 5, property used a*b = LCM * GCD
K and X have NO prime factors in common
1 Prime factor
3 Prime factors
Unauthorized copying prohibited
Apply in GMAT Context: Question 1
Does P have a factor X where1<X<P and X and P are positive integers?
1. GCD (P2, k) = k, where k is a prime number
2. 36*20 + 2 < P < 36*20+6
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Unauthorized copying prohibited
Question 1 – Steps 1 and 2 – Understand question statement and Draw Inferences
Question Statement - Does P have a factor X where1<X<P and X and P are positive integers?
Given –
X, P are positive integers
To find –
Does P have a factor X between 1 and P?
Is P Prime?
1 < X < P
P > 1
A prime number has only two factors: 1 and the number itself
A composite number will have at least one factor between 1 and the number itself
Think:
4: {1, 2, 4}
12: {1, 2, 3, 4, 6, 12}
Unauthorized copying prohibited
Question 1 – Step 3 – Analyze Statement 1 Independently
Statement 1: GCD (P2, k) = k, where k is a prime number
Statement 1 is not sufficient
X, P are positive integers Does P have a factor X between 1 and P?
Is P prime?
P2 is divisible by k
k is a prime number
k is a prime factor of P2 k is a prime factor of P
Xn has the same prime factors as X
P = nk
If n = 1P is Prime
If n ≠ 1P is NOT Prime
Think:Let k = 5 P = 5n
(n is an integer)
Unauthorized copying prohibited
Question 1 – Step 4 – Analyze Statement 2 Independently
Statement 2: 36*20 + 2 < P < 36*20+6
X, P are positive integers Does P have a factor X between 1 and P?
Is P prime?
Possible values of P:
36*20 + 3 36*20 + 4 36*20 + 5
3(12*20 + 1) 4(8*20 + 1) 5(36*4 + 1)
Divisible by 3 Divisible by 4 Divisible by 5
NOT Prime NOT Prime NOT Prime
Statement 2 is sufficient
Correct answer = Choice B = Statement 2 is sufficient but statement 1 is not
You need to be smart about simplifying information in order to arrive at the answer.
Unauthorized copying prohibited
Apply in GMAT Context: Question 2
If P and Q are positive integers and Q = 10 + 4P, find the GCD of P and Q
1. Q = 10 x, where x is a positive integer
2. P = 10 y, where y is a positive integer
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Unauthorized copying prohibited
Question 2 – Steps 1 and 2 – Understand question statement and Draw Inferences
Question Statement - If P and Q are positive integers and Q = 10 + 4P, find the GCD of P and Q
Given –
P, Q are positive integers
To find –
GCD of P and Q
Q = 10 + 4P
P = (Q- 10)/4
Unauthorized copying prohibited
Question 2 – Step 3 – Analyze Statement 1 Independently
Statement 1: Q = 10 x, where x is a positive integer
Statement 1 is not sufficient
Q = 10 + 4P P = (Q- 10)/4GCD of P and Q = ?P, Q are integers
P = (Q- 10)/4
P = (10x- 10)/4
P = 5(x- 1)/2P is an integer
x – 1 is even
ODD
x P Q GCD (P,Q)
3 5 30 5
5 10 50 10
7 15 70 5
GCD may be 5 or 10
Unauthorized copying prohibited
Question 2 – Step 4 – Analyze Statement 2 Independently
Statement 2: P = 10 y, where y is a positive integer
Statement 2 is sufficient
Q = 10 + 4P P = (Q- 10)/4GCD of P and Q = ?P, Q are integers
Q = 10 + 4P
Q = 10 + 4(10y)
Q = 10(4y + 1)
y P Q GCD (P,Q)
1 10 50 10
2 20 90 10
3 30 130 10
GCD is always10
Correct answer = Choice B = Statement 2 is sufficient but statement 1 is not
Final Words
Unauthorized copying prohibited
Key Takeaways
Know your concepts well.
Go through the concept files in detail and retain the basic and derived properties.
Follow the process with due diligence
Steps 1 & 2 – Understand the question and draw inferences
Step 3 – Analyze Statement 1
Step 4 – Analyze Statement 2
Practice how to simplify information in order to answer the question at hand
thoroughly completely
Decouple from statement 1
Will come with practice
Unauthorized copying prohibited
Next Steps
Analyze your performance in the Live
Session
Work on Weak Areas
◦ In Quant Online
◦ Number Properties block
Attempt Advanced Quiz
Prepare for next Session
◦ In Quant Online
◦ Divisibility and Remainders, Rounding, Statistics