### Important Formulas and Facts

Factors and Multiples:
If number x divided another number y exactly, we say that x is a factor of y.
and y is called a multiple of x.
Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.):
The H.C.F. of two or more than two numbers is the greatest number that divides them exactly.
Mehods To Find HCF
Factorization Method:
Step 1) Find the prime factors of each number
Step 2) Find the product of all common prime factors, which is the HCF of given numbers
e.g. To find the HCF of 14 and 8
14 = 2 x 7
8 = 2 x 2 x 2
HCF = 2
Division Method:
To find the H.C.F. of two given numbers
Step 1)Divide the larger by smaller one.
Step 2)Divide the divisor by the remainder.
Step 3)Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder.
The last divisor is the required H.C.F.
Finding the H.C.F. of more than two numbers:
Step 1)H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number.
Similarly, we can find the H.C.F. of more than three numbers.
Least Common Multiple (L.C.M.):
LCM is the least number which is exactly divisible by each one of the given numbers.
Methods to find LCM
Factorization Method:
Step 1)Find prime number factors of each given number.
Step 2)Find the product of highest powers of all the prime number factors, which gives the LCM of given numbers
e.g. To find the LCM of 14 and 8
14 = 2 x 7
8 = 2 X 2 x 2 = 2 3
LCM = 2 3 x 7 = 56
Division Method (short-cut):
Step 1)Arrange the given numbers in a rwo in any order.
Step 2)Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible.
Step 3)Repeat the above process till no two of the numbers are divisible by the same number except 1.
LCM = The product of the divisors and the undivided numbers
Product of two numbers = (HCF x L.C.M) of two numbers.
Co-primes: Co-primes means, if their H.C.F. is 1.
HCF and LCM of Fractions:
HCF = (HCF of Numerators) / (LCM of Denominators)
LCM = (LCM of Numerators) / (HCF of Denominators)
H.C.F. and L.C.M. of Decimal Fractions:
In a given numbers,
Step 1)Make the same number of decimal places by annexing zeros in some numbers, if necessary.
Step 2)Considering these numbers without decimal point, find H.C.F. or L.C.M.
Step 3)In the result, mark off as many decimal places as are there in each of the given numbers.
Comparison of Fractions:
Step 1)Find the L.C.M. of the denominators of the given fractions.
Step 2)Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

1)

252 can be expressed as a product of primes as :

A) 2 x 2 x 3 x 3 x 7

B) 2 x 2 x 2 x 3 x 7

C) 3 x 3 x 3 x 3 x 7

D) 2 x 3 x 3 x 3 x 7

2)

A, B and C start at the same time in the same direction to run around a circular stadium. A completes a round in 252 seconds, B in 308 seconds and C in 198 seconds, all starting at the same point. After what time will they meet again at the starting noint ?

A) 26 minutes 18 seconds

B) 42 minutes 36 seconds

C) 45 minutes

D) 46 minutes 12 seconds

3)

Four different electronic devices make a beep after every 30 minutes, 1 hour, 1.5 hour and 1 hour 45 minutes respectively. All the devices beeped together at 12 noon. They will again beep together at:

A) 12 midnight

B) 3 a.m.

C) 6 a.m

D) 9 a.m.

4)

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together ?

A) 4

B) 10

C) 15

D) 16

5)

Find the least number which when divided by 16, 18, 20 and 25 leaves 4 as remainder in each case, but when divided by 7 leaves no remainder.

A) 17004

B) 18000

C) 18002

D) 18004

6)

The smallest number which when diminished by 7, is divisible by 12, 16, 18 and 21 ?.

A) 1008

B) 1015

C) 1022

D) 1032

7)

Let the least number of six digits, which when divided by 4, 6, 10 and 15 leaves in each case the same remainder of 2, be N. The sum of the digits in N is :

A) 3

B) 4

C) 5

D) 6

8)

The least number which is a perfect square and is divisible by each of the numbers 10, 20 and 24, is

A) 1600

B) 3600

C) 6400

D) 14400

9)

The least number of five digits which is exactly divisible by 12, 15 and 18, is :

A) 10010

B) 10015

C) 10020

D) 10080

10)

Let N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case Then sum of the digits in N is :

A) 4

B) 5

C) 6

D) 8

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