Problem Solving

M7MBA wrote: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

Since W divided by 4,233 yields a remainder of 61, W could equal 4,233 + 61 = 4,294.

Thus, 2W = 2 x 4,294 = 8,588.

We can now divide 8,588 by 83 and we have:

8,588/83 = 103 remainder 39

Alternate Solution: W divided by 4233 produces a remainder of 61 means that W = 4233q + 61 for some integer q. Multiplying each side of the equation by 2, we obtain 2W = 8466q + 122. Note that 83 divides evenly into 8466 (because 8466 = 83 x 102) and that 122 = 83 + 39. Then, 2W = 83(102)q + 83 + 39 = 83(102q + 1) + 39. Since 83 divides evenly into 83(102q+1), the remainder is 39.

Answer: C

M7MBA wrote: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

IMPORTANT: 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.

So, rather than test much bigger values of W, let's see what happens when W = 61
If W = 61, then 2W = 2(61) = 122

What is the remainder when 2W is divided by 83?
83 divides into 122 one time with remainder 39

Answer: C

Cheers,
Brent

M7MBA wrote: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

The OA is C .

How can I know which one is the correct answer? Experts, I need your help, please.

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

Aside: 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

Cheers,
Brent