Session Numbers

Factors – II

Numbers is one of the most important topics for CAT and other management entrance exams, questionsfrom which have appeared consistently and in significant numbers in all these exams.

Key concepts discussed:

• Factors (divisors) of a number are the numbers which divide the number completely.• Prime factorization is the process of expressing a composite number as a product of its prime

factor(s).

• If p q rN a b c ...= × × × , where a, b, c,… are prime numbers such that a < b < c …, then

o Number of factors of N = (p + 1) × (q + 1) × (r + 1) × …;

o Number of even factors of N = p × (q + 1) × (r + 1) × … if a = 2;

o Number of even factors of N = 0 if a ≠ 2;

o Number of odd factors of N = 1 × (q + 1) × (r + 1) × …;

o Number of ways of expressing N as a product of two factors (Np) = (p 1) (q 1) (r 1) ...

2+ × + × + ×

(when N is not a perfect square);

o Number of ways of expressing N as a product of two factors (Np) = (p 1) (q 1) (r 1) ... 1

2+ × + × + × +

(when N is a perfect square);

o Number of ways of expressing N as a product of two distinct factors

( )p'(p 1) (q 1) (r 1) ... 1

N2

+ × + × + × −= (when N is a perfect square);

o Number of ways of expressing N as a product of two co-prime factors = 2n – 1 (where n is thenumber of prime factors of the number);

o Product of factors of N (N is not a perfect square) = NpN ;

o Product of factors of N (N is a perfect square) = 1

Np 2N−

;

o Sum of factors of N = (a0 + a1 + a2 + … + ap) × (b0 + b1 + b2 + … + bq) × (c0 + c1 + c2 +…+ cr) ×…;

o Sum of even factors of N = (a1 + a2 + … + ap) × (b0 + b1 + b2 + … + bq) × (c0 + c1 + c2 +…+ cr) ×… (where a = 2);

o Sum of odd factors of N = a0 × (b0 + b1 + b2 + … + bq) × (c0 + c1 + c2 +…+ cr) × … (where a = 2).

Highlight: This session deals with definition and formulae based questions which are of moderate difficultylevel.

SessionNumbers

The questions discussed in the session are given below along with their source.

Q1. If the area of a rectangular plot is 4400 square meters, then how many integral dimensions of theplot are possible whose breadths are at least 20 m?(a) 8 (b) 7 (c) 9 (d) 11

(CL material)

Instructions: Consider the information given below for questions 2 and 3.In the diagram below, the seven letters correspond to seven unique digits chosen from 0 to 9. The relationshipamong the digits is such that:

P

Q

R

A

X

Y

Z

P.Q.R = X.Y.Z = Q.A.Y

Q2. The value of A is:(a) 0 (b) 2 (c) 3 (d) 6 (e) None of the above.

(XAT 2009)

Q3. Let X be a four-digit number with exactly three consecutive digits being same and is a multiple of 9.How many such X’s are possible?(a) 12 (b) 16 (c) 19 (d) 21 (e) None of the above.

(XAT 2009)

Q4. Find the product of odd factors of 360.(a) 2025 (b) 91125 (c) 216125 (d) None of these

(CL material)

Question No. 5 are followed by two statements labelled as I and II. Decide if these statements aresufficient to conclusively answer the question. Choose the appropriate answer from the optionsgiven below:

A. Statement I alone is sufficient to answer the question.B. Statement II alone is sufficient to answer the question.C. Statement I and Statement II together are sufficient, but neither of the two alone is

sufficient to answer the question.D. Either Statement I or Statement II alone is sufficient to answer the question.E. Neither Statement I nor Statement II is necessary to answer the question.

Q5. Given below is an equation where the letters represent digits.(PQ). (RQ) = XXX. Determine the sum of P + Q + R + X.I. X = 9.II. The digits are unique.

(XAT 2011)

Session Numbers

Errata:Despite the best of our efforts, minor slips have crept into this session which are elaboratedbelow for your convenience.

In question no. 2, at 10:59, GP mistakenly marks the answer as option (d) 6, in place of option

(b) 2. The value of A is 72

24 9

=× .