m and n are positive integers, each greater than 1. If 7(m–1)=5(n–1),

Question

m and n are positive integers, each greater than 1. If 7(m – 1) = 5(n – 1), which of the following options could be true?

A. m + n = 14

B. m – n = –2

C. n – m = –2

A. I only 
B. II only
C. III only
D. I and II only
E. I, II, and III

Ravi1410 · Answer

We can substitute value of m from all the 3 options and see whether we are getting positive integer for n.

I. m+n=14 => m=14-n
7(14-n-1) = 5(n-1)
91 - 7n = 5n -5
96 = 12n
n = 8 and m = 6. It satisfies the condition both are positive integers.

II. m-n = -2 => m = n-2
7(n-2-1) = 5(n-1)
7n - 21 = 5n-5
2n = 16
n = 8 and m = 6. It satisfies the condition both are positive integers.

III. n-m = -2 => m = n+2
7(n+2-1) = 5(n-1)
7n+7 = 5n-5
2n = -12
n=-6 and m = -4. It fails the condition that m and n are positive integers.

So only option I and II are correct.

Ans: Option D

Kinshook · Answer

Given: m and n are positive integers, each greater than 1. 
Asked: If 7(m – 1) = 5(n – 1), which of the following options could be true?

(m-1)/(n-1) = 5/7
n>m
(m,n)= {(6,8),(11,15)

A. m + n = 14
m + n = 6 + 8 = 14; Could be true
B. m – n = –2
m-n = 6-8 = -2; Could be true
C. n – m = –2
Since n>m; n-m>0; Can not be true

IMO D

ScottTargetTestPrep · Answer

Solution:

We can create a ratio:

7(m - 1) = 5(n - 1)

(m - 1) / (n - 1) = 5/7

We see that n > m. We can also see that m = 6 and n = 8 isone possible solution for the values of m and n.

Let’s substitute m = 6 and n = 8 into the three statements:

A. m + n = 14
Our values of m = 6 and n = 8 show that A could be true.

B. m – n = –2
Again, our values of m = 6 and n = 8 show that B could be true.

C. n - m = -2
We earlier showed that n > m, so the difference n - m could never be negative.

Answer: D

MathRevolution · Answer

7(m – 1) = 5(n – 1)

This is possible when 7 * 5 = 5 * 7

Therefore, m - 1 = 5 and n - 1 = 7

=> m - 1 = 5 => m = 6

=> n - 1 = 7 => n = 8

=> m + n = 6 + 8 = 14   - OPTION I

=> m - n = 6 - 8 = -2   - OPTION II

=> n - m = 8 - 6 = 2   - NO OPTION

Only Options I and II are TRUE.

Answer D

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