arithmetic lcm and hcf
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Number SystemsNatural & Whole numbers
Integers & FractionsRational & Irrational numbers
Prime & Composite numbers
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Divisibility Rules
A number is divisible by 2 if
That number is an evennumber (the last digit of the
number is an even number)
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Divisibility Rules
A number is divisible by 4 if
The last two digits of that
number are divisible by 4
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Divisibility Rules
A number is divisible by 5 if
The last digit is 0 or 5.
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Divisibility Rules
A number is divisible by 6 if
That number is divisible byboth 3 and 2 (an even number,
whose digits add up to anumber divisible by 3)
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Divisibility Rules
A number is divisible by 8 if
The last three digits of that
number are divisible by 8
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Divisibility Rules
A number is divisible by 9 if
The sum of all the digits of the
number is divisible by 9.
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Divisibility Rules
A number is divisible by 11 if
The difference of the Sum of
alternate digits to be 0 ordivisible by 11
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Addition and Multiplication rulesfor Even and Odd numbersEven + Even = Even
Even * Even = EvenEven + Odd = Odd
Even * Odd = EvenOdd + Odd = Even
Odd * Odd = Odd
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Can Even / Odd be aninteger?
Depends on the numbers.
For example, 28/7 is an integer.
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Can Odd / Odd be aneven integer?
Never
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Divide an even number with
an odd number. If remainderexists, is it odd or even?
18/5 - Remainder is 3 - odd
30/ 7 - Remainder is 2 - evenNothing can be said.
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Indices
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OddOdd= ?
Odd
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EvenOdd= ?
Even
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a3* a2= ?
a5
Rule: am* an= a m+n
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(a3)2= ?
a6
Rule: (am) n= a m n
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a3/a2= ?
aRule: am/ an= a m - n
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(ab)3= ?
a3 b3
Rule: (ab)m= a m bm
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a- 3 = ?
1/ a3
Rule: (a) - m= 1 / a m
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What if a is a fraction or
a negative number?
The rules are still applicable
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LCM / HCF
Wh t th lti l f
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What are the multiples of
6 and 9?6: 6, 12, 18, 24, 30
9: 9, 18, 27, 36
Common Multiples: 18, 36, ..
LCM: 18
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What are the factors of 6
and 9?6: 1, 2, 3, 6
9: 1, 3, 9
Common Factors: 1, 3
HCF: 3
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Find the LCM and HCF of2835 & 3456.
LCM = 362880, HCF = 27
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Find the LCM and HCF of
3/4 and 1/2.
LCM = 3/2, HCF = 1/4
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The LCM of two fractions
a/b and c/d is
(LCM of a & c) / (HCF of b & d)
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The HCF of two fractions
a/b and c/d is
(HCF of a & c) / (LCM of b & d)
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Exercise
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What is the fraction form
of 0.33333....?
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What is the fraction form
of 0.44444.. ?
4/9
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What is the fraction form
of 0.12121212.. ?
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What is the last digit of71*72*73*...*79?
The product of 72 and 75 willend in a zero.
So, the last digit of the productwill be zero.
d b th di it f
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a and b are the digits of a
two digit number. Then(ab - ba) is always a
multiple of ...ab = 10a + b, ba = 10b + a
ab - ba = 9a - 9b
It will always be a multiple of 9
f f
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A sum of five successive
whole numbers is alwaysdivisible by ...
Let the numbers be x, x+1, x+2,
x+3, x+4. Their sum is 5x + 10.It will always be divisible by 5.
When divided by 8 any (odd
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When divided by 8, any (oddnumber)2gives ___ as a
remainder
Square of odd = (2a+1)2= 4a 2+ 4a + 1 = 4 (a 2+a) + 1
(a 2+a) will be even, for all a.
4(a 2+a) will be multiple of 8
Th d t
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The product
67
x 353
x 1110
has howmany prime factors?
6 has 2 prime factors - 2 & 3So 67has 14 prime factors
Similarly 353has 6 and
1110has 10 prime factors.
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Mentally calculate the
square of 65
Take the tens digit, 6, add 1to it.
Multiply the two 6*7 = 42Append 25 to this - 4225.
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Mentally calculate 58*52Take the tens digit, 5, add 1 to
it.
Multiply the two 6*5 = 30Append 8*2 = 16 to this - 3016.
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What is 245 x 25?
= (250*25) - (5*25)
= 6250 - 125= 6125
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What is 204?
142= 196
Calculate (14.1)2, (14.2)2andso on (algebraically).
So the square root is between14 and 15, but closer to 14.
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What is 66% of 999 ?
= approx. 66% of 1000
= 660
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Which is larger 103*97 or
107*93?
103*97 = 1002
- 32
107*93 = 1002- 72
So 103*97 is bigger
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Which is greater - 16/66
or 25%?
16/66 < 16/64So, the fraction is less than 1/4,
or less than 25%So 25%is bigger
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What is 107*107?
= (100 +7)2
= 1002+ 2*100*7 + 72
= 10,000 + 1400 + 49 = 11449
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Which is bigger 999- 998
or 998?
999- 998= 998(9-1)
= 8*998
So 999- 998is bigger.
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If I first your marks by
10% and then them by10%, which are more:your earlier marks or
changed marks?The earlier marks are more.
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What is the remainder
when 1012+ 5 is dividedby 3?
The sum of digits of 1012+5 is 6.
Since this is divisible by 3,
the remainder is 0.
I 5353 3333 di i ibl b
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Is 5353- 3333divisible by
10?Last digits of a number ending
with 3 will be..31= 3 32= 9
33= 7 34= 1
.
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After every 4thpower the series
will repeat.
So 353and 333will both end in
3.
And the difference will be
divisible by 10.
3 b d i l
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3 boys go round a circular
field in 3, 4 and 5 minutesrespectively. They start from
the same place at the sametime. When will they be at
the same place again?They will meet after 60 minutes
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Revision
C i f i
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Conversion of recurring
decimal 0.abcabcabc .. tofraction is
abc / 999
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Shortcut for square of a5?
a(a+1),25
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LCM * HCF =
Product of the two numbers
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The LCM of two fractions
a/b and c/d is
(LCM of a & c) / (HCF of b & d)
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The HCF of two fractions
a/b and c/d is
(HCF of a & c) / (LCM of b & d)
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Averages
A t i f h
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A countrys airforce has
18 squadrons and a totalof 558 aircraft. What is the
average number of aircraftin a squadron?
Average = 558 / 18 = 31
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Average
Total of class________________________
number of elements in class
If th l i
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If the annual p.c. income
in X is $130 and that in Yis $90, what is the
average p.c. income of Xand Y taken together?
= 130 + 90 / 2 = 220 / 2 = 110
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If it is given that Ys
population is half of Xs?= (130* 2x + 90 * x) / 3x =116.7
If ith 4 t d
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If a car with 4 tyres and a
spare traveled 20000 km,and all tyres have been
used equally, what wasthe mileage of each tyre?
80000 / 5 = 16000
Which is the smallest
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Which is the smallest
whole number, whenmultiplied with 576, makes
a perfect cube?576 = (23) * (23) * (32)
Multiply by 3 to get a perfect
cube
What should be minimum
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What should be minimum
number of boys so thatthey can be arranged in
the group of 20, 25, 35 or40 ?
The LCM of 20, 25, 35 and 40
= 1400.
A batsman's avg for
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A batsman s avg. for
some innings was 42.5.Next 4 innings he scored
45, 48, 41 & 56, and hisavg. for all the innings
became 43.75. How manyinnings in all did he play ?
Total till start = 42 5x
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Total till start = 42.5x.New = 42.5x + (45+48+41+56).Avg. = (42.5x + 190)/ (x+4) =43.75
Solve for x, answer = 12.
The average age of Ajit
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The average age of Ajit
and his wife when theymarried 5 years ago was
24. What is the presentaverage age of the family
if they have a 2 year-oldson?
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The couples current age =
2(24 + 5) = 58.Including son, total = 60.
Average = 60/3 = 20.
Jill earned $ 1300 last
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Jill earned $ 1300 last
month. If her basic salaryis $ 1000 more than her
overtime, how much didshe earn in overtime?
Let overtime be x;x + (1000+x) = 1300. x = 150.
A number is created by
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A number is created by
writing the first 54numbers:
12345678............. 5354.This, when divided by 8,
will leave a remainder of?Remainder will be 354-352 = 2
What is the number if
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What is the number, if
increased by 7, issimultaneously divisible
by 27, 21, 35 and 77?
LCM of above numbers =21*99*5 = 10395
number = 10395 7 = 10388
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Mr X enters a positive
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Mr. X enters a positive
number y in a calculatorand then goes on
pressing the square root() key repeatedly. What
will happen?
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For any positive number, the
larger the n, the closer thevalue of nthroot to 1.
The product of all
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The product of all
integers from 1 to 100 willhave -- number of zeroes
at the end.One each for each multiple of 5from 5 to 100; four more for 25,50, 75, and 100. Answer = 24.
Three wheels rotate at 60
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Three wheels rotate at 60,
36 and 24 rpm. Each hasa red spot on its rim,
which coincides at thestart. The red spots will be
at this position again afterhow many seconds?
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The wheels make 1, 0.6 and
0.4 rev/sec respectively.LCM of the three is 6 sec.
Th LCM f t b
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The LCM of two numbers
is 12 times their HCF. Thesum of the HCF and LCM
is 403. If one of thenumbers is 93, find theother.
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Let HCF = x. LCM = 12x.
LCM+HCF =13x = 403.
So x = 31
LCM*HCF = Product
Other number = 12*31*31 / 93= 124
A, B & C sit together for a
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A, B & C sit together for a
meal. A has 3 loaves, Bhas 4 and C has none.
They share the loavesequally and C pays $21.
How much should B get?
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Loaves eaten/person = 7/3,A contributed 3 - 7/3, = 2/3,
and B contributed 4 - 7/3 = 5/3.So, B gets 5/7 * 21 = $15
X engaged Y at a salary
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X engaged Y at a salaryof $ 2400 for 30 days. Yforfeits $ 100/day that he
didnt work. At the end,neither owed the other
any money. How manydays did Y work?
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80x = 100(30-x).
So x = 3000/180=16 and 2/3days
A two digit number "ab"
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A two digit number abis squared. The square ofthis number is of the form
"ccb". If we know that"ccb" is greater than 300,
then what is the value ofb?
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172= 289. 352> 1000.
So, the number is between 17and 35.
"b" can take only values0,1,5,6.
The only number that fits is 21.
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