LCM and HCF Formulas

LCM by Prime Factorization Method

A technique to find the Least Common Multiple (LCM) of a set of numbers by breaking down each number into its prime factors and then multiplying the highest powers of each prime factor.

Lets take two numbers i.e., 25 and 35, now to calculate the LCM:

• List the prime factors of each number first.
  25 = 5 × 5
  35 = 7 × 5

• Then multiply each factor the most number of times it occurs in any number.

If the same multiple occurs more than once in both the given numbers, then multiply the factor by the most number of times it occurs.
The occurrence of Numbers in the above example:
5: two times
7: one time
LCM = 7 × 5 × 5 = 175

LCM by Division Method

Let us see with the same example, which we used to find the LCM using prime factorization.
Solve LCM of (25,35) by division method.

5 | 25, 35
----------
5 | 5, 7
---------
7 | 1, 7
---------
  | 1, 1

Therefore, LCM of 25 and 35 = 5  x 5 × 7 = 175

Questions and Answers of HCF and LCM

Question:

Calculate the highest number that will divide 43, 91 and 183 and leaves the same remainder in each case

Options

A. 4
B. 7
C. 9
D. 13

Solution:   

Here the trick is :

1. Find the Differences between numbers
2. Get the HCF ( that differences)

We have here 43, 91 and 183

So differences are

183 – 91 = 92,
183 – 43 = 140,
91 – 43 = 48.

Now, HCF (48, 92 and 140)

• 48 = 2 × 2 × 2 × 2 × 3
• 92 = 2 × 2 × 23
• 140 = 2 × 2 × 5 × 7
• HCF = 2 × 2 = 4

And 4 is the required number.

Correct Answer : A

Question:

Which of the following is  greatest number of four digits which is divisible by 15, 25, 40 and 75 is:

Options

A. 9700
B. 9600
C. 9800
D. 9650

Solution:    Greatest number of 4-digits is 9999.

Now , find the L.C.M. of 15, 25, 40 and 75 i.e.  600.

On dividing 9999 by 600, the remainder is 399.

Hence,  Required number (9999 – 399) = 9600.

Alternatively, 

\frac{9999}{600} = 16.66500

Ignore the decimal points, required number would be 16 * 600 = 9600

Correct Answer : B

Question: 

The greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm, 12 m 95 cm is:

Options

A. 25 cm
B. 15 cm
C. 35 cm
D. 55 cm

Solution:     Required length = H.C.F. of 700 cm, 385 cm and 1295 cm = 35 cm.

Correct Answer : C