Find the L.C.M. of the following numbers by division method. a) 9 and 12 b) 12 and 15 c) 18 and 24 d) 30 and 45 e) 12, 15, 20 f) 20, 30, 40 g) 25, 30, 75 h) 20, 36, 45
The Correct Answer and Explanation is:
To find the Least Common Multiple (L.C.M.) of the given numbers using the division method, we will follow these steps:
Step-by-Step Process:
- Write the numbers in a row.
- Start dividing them by the smallest prime number that divides at least one of the numbers.
- Continue dividing until no further division is possible, then multiply the divisors to get the L.C.M.
Solution:
a) L.C.M. of 9 and 12:
- Numbers: 9, 12
- Divide by 3:
- 9 ÷ 3 = 3
- 12 ÷ 3 = 4
- Divide by 3 again:
- 3 ÷ 3 = 1
- 4 ÷ 3 (not divisible)
- Divide by 2:
- 4 ÷ 2 = 2
- Divide by 2 again:
- 2 ÷ 2 = 1
- Remaining divisors: 3, 3, 2, 2
- L.C.M = 3 × 3 × 2 × 2 = 36
b) L.C.M. of 12 and 15:
- Numbers: 12, 15
- Divide by 3:
- 12 ÷ 3 = 4
- 15 ÷ 3 = 5
- Divide by 5:
- 4 ÷ 5 (not divisible)
- 5 ÷ 5 = 1
- Divide by 2:
- 4 ÷ 2 = 2
- Divide by 2 again:
- 2 ÷ 2 = 1
- Remaining divisors: 3, 5, 2, 2
- L.C.M = 3 × 5 × 2 × 2 = 60
c) L.C.M. of 18 and 24:
- Numbers: 18, 24
- Divide by 2:
- 18 ÷ 2 = 9
- 24 ÷ 2 = 12
- Divide by 2 again:
- 9 ÷ 2 (not divisible)
- 12 ÷ 2 = 6
- Divide by 3:
- 9 ÷ 3 = 3
- 6 ÷ 3 = 2
- Divide by 2:
- 3 ÷ 2 (not divisible)
- 2 ÷ 2 = 1
- Remaining divisors: 2, 2, 3, 3
- L.C.M = 2 × 2 × 3 × 3 = 72
d) L.C.M. of 30 and 45:
- Numbers: 30, 45
- Divide by 3:
- 30 ÷ 3 = 10
- 45 ÷ 3 = 15
- Divide by 3 again:
- 10 ÷ 3 (not divisible)
- 15 ÷ 3 = 5
- Divide by 5:
- 10 ÷ 5 = 2
- 5 ÷ 5 = 1
- Remaining divisors: 3, 3, 5
- L.C.M = 3 × 3 × 5 × 2 = 90
e) L.C.M. of 12, 15, and 20:
- Numbers: 12, 15, 20
- Divide by 2:
- 12 ÷ 2 = 6
- 15 ÷ 2 (not divisible)
- 20 ÷ 2 = 10
- Divide by 2 again:
- 6 ÷ 2 = 3
- 10 ÷ 2 = 5
- Divide by 3:
- 3 ÷ 3 = 1
- 5 ÷ 3 (not divisible)
- Divide by 5:
- 5 ÷ 5 = 1
- Remaining divisors: 2, 2, 3, 5
- L.C.M = 2 × 2 × 3 × 5 = 60
f) L.C.M. of 20, 30, and 40:
- Numbers: 20, 30, 40
- Divide by 2:
- 20 ÷ 2 = 10
- 30 ÷ 2 = 15
- 40 ÷ 2 = 20
- Divide by 2 again:
- 10 ÷ 2 = 5
- 15 ÷ 2 (not divisible)
- 20 ÷ 2 = 10
- Divide by 5:
- 5 ÷ 5 = 1
- 10 ÷ 5 = 2
- Divide by 2:
- 2 ÷ 2 = 1
- Remaining divisors: 2, 2, 5, 5
- L.C.M = 2 × 2 × 5 × 5 = 100
g) L.C.M. of 25, 30, and 75:
- Numbers: 25, 30, 75
- Divide by 5:
- 25 ÷ 5 = 5
- 30 ÷ 5 = 6
- 75 ÷ 5 = 15
- Divide by 5 again:
- 5 ÷ 5 = 1
- 6 ÷ 5 (not divisible)
- 15 ÷ 5 = 3
- Divide by 3:
- 3 ÷ 3 = 1
- 6 ÷ 3 = 2
- Divide by 2:
- 2 ÷ 2 = 1
- Remaining divisors: 5, 5, 3, 2
- L.C.M = 5 × 5 × 3 × 2 = 150
h) L.C.M. of 20, 36, and 45:
- Numbers: 20, 36, 45
- Divide by 2:
- 20 ÷ 2 = 10
- 36 ÷ 2 = 18
- 45 ÷ 2 (not divisible)
- Divide by 2 again:
- 10 ÷ 2 = 5
- 18 ÷ 2 = 9
- Divide by 3:
- 5 ÷ 3 (not divisible)
- 9 ÷ 3 = 3
- Divide by 3 again:
- 3 ÷ 3 = 1
- Remaining divisors: 2, 2, 3, 3, 5
- L.C.M = 2 × 2 × 3 × 3 × 5 = 180
Summary of Results:
- a) 36
- b) 60
- c) 72
- d) 90
- e) 60
- f) 100
- g) 150
- h) 180
