M40-05

Question

When the positive integers  m  and  n  are divided by the positive integer  d , they yield remainders of 17 and 27, respectively. If  m + n , when divided by  d , results in a remainder of 7, what is the value of  d ?

A. 34
B. 37
C. 44
D. 47
E. 51

Bunuel · Answer

Official Solution:

When the positive integers  m  and  n  are divided by the positive integer  d , they yield remainders of 17 and 27, respectively. If  m + n , when divided by  d , results in a remainder of 7, what is the value of  d ?

A. 34
B. 37
C. 44
D. 47
E. 51

The information given in the problem can be expressed as follows:
   
 m = dq + 17 , where  d > 17 .
 
 n = dp + 27 , where  d > 27 .
   
Adding these equations gives:
   
 m + n = d(q+p) + 44 .
   
Given that  m + n , when divided by  d , yields a remainder of 7, we can rewrite the equation as:
   
 m + n = d(q+p) + 37 + 7 .
   
This implies that  d  must be a factor of 37, and since  d > 27 , the value of  d  must be 37.

Answer: B

Raido7 · Answer

Here's a slightly different way of thinking about it -

from the division algorithm we get : 
dividend = (divisor * quotient) + remainder

m = dQ + 17 --------- (1)
n = dP + 27 -----------(2)
m+n = dR + 7 -----(3)

substituting m + n we get

dQ + dP + 44 = dR + 7
=> d (R - (Q+P)) = 37
=> d = 37/ (R-(Q+P))

since R,Q and P are quotients, they must be integers. Hence the only way we can get d as one of the options is if (R - (Q+P)) = 1

Therefore d= 37 / 1 i.e  d = 37

MBAToronto2024 · Answer

KarishmaB 
I got the following solution: 
m = kd + 17 
n = kd + 27
m + n = 2kd + 44
2kd + 44 = kd + 7
kd = -37 
Not sure if it's correct but looks like that. And I selected 37 as answer.

KarishmaB · Answer

Note that the quotients in the two cases are likely to be different so we shouldn't use k for all.

m = kd + 17
n = td + 27

m + n = d(k+t) + 44

d(k+t) + 44 = ds + 7

d(s-k-t) = 37*1

So d must be 37 and (s-k-t) = 1

Conceptual Method:

Instead, think about it in terms of division as grouping.

When m is divided by d, some groups of d balls are formed and 17 balls remain.
When n is divided by d, some groups of d balls are formed and 27 balls remain. 
When m+n is divided by d, the groups of d balls come together and so do 17 + 27 balls = 44 balls.

But actually only 7 balls remain. Why? Because another group of 37 is formed from 44 and 7 balls are left. Hence d must be 37.

Answer (B)

kubrah · Answer

Following your logic, would it be fair to say that for these types of problems the answer is simply to sum the remainders then subtract the remainder thats left over at the end?

Thanks for your explanation by the way! Appreciate your posts.

braddouglas10 · Answer

O great math wizard, Bunuel, do you have any more problems similar to this one?

Bunuel · Answer

braddouglas10 · Answer

Thank you, great math wizard, Bunuel!

phmahi1997 · Answer

I like the solution - it’s helpful.

phmahi1997 · Answer

I like the solution - it’s helpful.

phmahi1997 · Answer

Another Amazing question on reminder! I was practicing a reminder problem set; my concept is now sealed with these fantastic and  high-quality  questions! Thanks!

juliansorre · Answer

Another way to solve it:

d cannot be a multiple of 17, 27, or 44, so one can discard 34, 44 and 51. 
Furthermore, d must be less than 44. So the only alternative is 37.

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