Arithmetic LCM

8/8/2019 Arithmetic LCM

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Arithmetic


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Number Systems

Natural & Whole numbers

Integers & Fractions Rational & 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


The sum ofall the digits of the

number is divisible by 3.


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Divisibility Rules


The last two digits of that

numberare divisible by 4


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Divisibility Rules


The last digit is 0 or 5.


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Divisibility Rules


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


The last three digits of that

numberare divisible by 8


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Divisibility Rules


The sum ofall the digits of the

number is divisible by 9.


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Divisibility Rules


The difference of the Sum of

alternate digits to be 0 ordivisible by 11


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Addition and Multiplication rules

for Even and Odd numbers Even + Even = Even

Even * Even = Even Even + Odd = Odd

Even * Odd = Even Odd + 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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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 = ?

a

6

Rule: (am) n = a m n


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a3/a2 = ?

a

Rule: am/ an = a m - n


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(ab)3 = ?

a3 b3

Rule: (ab)m = a m b m


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a- 3 = ?

1/ a3

Rule: (a) - m = 1 / a m


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Wha

t if a

isa

fra

ction ora negative number?

The rules are stillapplicable


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LCM / HCF


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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 theLCM

a

nd HCF of3/4 and 1/2.

LCM = 3/2, HCF = 1/4


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The LCM of two fractionsa/b and c/d is

(LCM ofa & c) / (HCF of b & d)


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The HCF of two fractionsa/b and c/d is

(HCF ofa & c) / (LCM of b & d)


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Exercise


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What is the fraction formof 0.33333....?

1/3


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What is the fraction formof 0.44444.. ?

4/9


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What is the fraction formof 0.12121212.. ?

12/99


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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.


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aand b are the digits ofa

two digit number. Then(ab - ba) is always a

multiple of ...ab = 10a + b, ba = 10b + a

ab - ba = 9a - 9b

It willalways be a multiple of 9


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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 willalways be divisible by 5.


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When divided by 8, any (odd

number)

2

gives ___asaremainder

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


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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 353 has 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*52

Tak

e the tens digit, 5,add 1 toit.

Mul

tiply the two 6*5 = 30

Append 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)2 andso on (algebraically).

So the square root is between14 and 15, but closer to 14.


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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 - 16/66or 25%?

16/66 < 16/64So, the fraction is less than 1/4,

orless 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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If I first o your marks by

10% and then q them by10%, which are more:your earlier marks or

changed m

arks?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.


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After every 4

th

power the serieswill repeat.

So 3

53

and 3

33

will both end in3.

And the difference will bedivisible by 10.


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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


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Conversion of recurring

decimal 0.abcabcabc .. tofraction is

abc / 999


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Shortcut for square ofa5?

a(a+1),25


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LCM * HCF =

Product of the two numbers


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The LCM of two fractionsa/b and c/d is

(LCM ofa & c) / (HCF of b & d)


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The HCF of two fractionsa/b and c/d is

(HCF ofa & c) / (LCM of b & d)


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Averages


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A countrys airforce has

18 squadrons and a totalof 558 aircraft. What is the

average number ofaircraftin a squadron?

Average = 558 / 18 = 31


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Average

Total of class________________________

number of elements in class


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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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Ifa 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

Whi h i th ll t


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Which is the smallest

whole number, whenmultiplied with 576, makes

a perfect cube?

576 = (2

3

) * (2

3

) * (3

2

)Multiply by 3 to get a perfectcube.

Wh t h ld b i i


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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 b t ' f


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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. forall the innings

became 43.75. How manyinnings in all did he play ?

T t l till t t 42 5


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Total till start = 42.5x.N

ew = 42.5x + (45+48+41+56).Avg. = (42.5x + 190)/ (x+4) =43.75Solve for x, answer = 12.


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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.


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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 ofabove n

umbers =

21*99*5 = 10395

number = 10395 - 7 = 10388

X had a batting average


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X had a batting average

of 40.8/match in 15 one-day matches. Later, it was

discovered that, in the 2 ofthose matches, his score

was 48 and not 84. Hiscorrect average is ...


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[ 15 * 40.8 - (84-48)*2 ] / 15 =36

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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Forany positive number, thelarger the n, the closer thevalue of nth root to 1.

The product of all


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The product ofall

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.


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The LCM of two numbers

is 12 times their HCF. Thesum of the HCF and LCMis 403. If one of the

nu

mbers 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 fora

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 salary

of $ 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 2/3 days

A two digit number "ab"


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A two digit number ab

is 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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Arithmetic LCM