Suppose 72 = m x n, where m and n are positive integers such that 1 < m < n. How many possible values of m are there?
[amp_mcq option1="5" option2="6" option3="10" option4="12" correct="option1"]
This question was previously asked in
UPSC CAPF – 2018
There are 5 possible values for m.
– We are given that 72 = m x n, where m and n are positive integers such that 1 < m < n.
- We need to find pairs of factors (m, n) of 72 that satisfy the condition 1 < m < n.
- First, list pairs of factors (m, n) of 72 such that m <= n:
- 72 = 1 x 72 (m=1, n=72)
- 72 = 2 x 36 (m=2, n=36)
- 72 = 3 x 24 (m=3, n=24)
- 72 = 4 x 18 (m=4, n=18)
- 72 = 6 x 12 (m=6, n=12)
- 72 = 8 x 9 (m=8, n=9)
- Now, apply the condition 1 < m < n:
- (1, 72): m=1. Fails 1 < m.
- (2, 36): m=2. Satisfies 1 < 2 < 36. m=2 is a possible value.
- (3, 24): m=3. Satisfies 1 < 3 < 24. m=3 is a possible value.
- (4, 18): m=4. Satisfies 1 < 4 < 18. m=4 is a possible value.
- (6, 12): m=6. Satisfies 1 < 6 < 12. m=6 is a possible value.
- (8, 9): m=8. Satisfies 1 < 8 < 9. m=8 is a possible value.
- The possible values for m are the first elements of the valid pairs: 2, 3, 4, 6, 8.
- There are 5 possible values for m.