Problem Solving

A remainder question I just made up.
Estimated difficulty level: 650

When positive integer W is divided by 4233, the remainder is 61. What is the remainder when 2W is divided by 83?

A) 11
B) 12
C) 39
D) 44
E) 61

Answer: C

To further David's point, when it comes to remainders, we have a nice rule that says:

If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.

For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

So, for this question, we're told that, when positive integer W is divided by 4233, the remainder is 61.
This means the possible values of W are: 61, 61 + 4233, 61 + (2)(4233), 61 + (3)(4233), 61 + (4)(4233), . . . etc.

Cheers,
Brent

Brent@GMATPrepNow wrote:A remainder question I just made up.
Estimated difficulty level: 650

When positive integer W is divided by 4233, the remainder is 61. What is the remainder when 2W is divided by 83?

A) 11
B) 12
C) 39
D) 44
E) 61

Answer: C

Another (much longer) solution involves another important rule regarding remainders:

"If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

We'll also use the fact that 8466 = (83)(102)

When positive integer W is divided by 4233, the remainder is 61
So, we can say W = 4233Q + 61 for some integer Q.

This means that...
2W = 2(4233Q + 61)
= 8466Q + 122
= (83)(102)Q + 122
= (83)(102)Q + 83 + 39
= 83(102Q + 1) + 39

In other words, 2W is 39 greater than some multiple of 83.
So, when we divide 2W by 83, the remainder will be 39

Answer: C