The integers x and y are both positive, the remainder when

BTGModeratorVI wrote: ↑

Tue May 12, 2020 2:18 pm

The integers x and y are both positive, the remainder when x is divided by 12 is 7, and the remainder when y is divided by 12 is 3. Each of the following is a possible value of 2x+y EXCEPT

A. 125
B. 101
C. 77
D. 51
E. 41

Answer: D
Source: Kaplan

One approach is to test values

IMPORTANT: 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.

Okay, onto the question....

The remainder when x is divided by 12 is 7
Possible values of x = 7, 19, 31, 43, 55, ...
So, possible values of 2x are: 14, 38, 62, 86, 110, ...

The remainder when y is divided by 12 is 3
Possible values of y are: 3, 15, 28, 40, 52, ...

Each of the following is a possible value of 2x + y EXCEPT

A) 125 This equals 110 + 15. ELIMINATE A
B) 101 This equals 86 + 15. ELIMINATE B
C) 77 This equals 62 + 15. ELIMINATE C
D) 51
E) 41 This equals 38 + 3. ELIMINATE E

Answer: D

Cheers,
Brent

BTGModeratorVI wrote: ↑

Tue May 12, 2020 2:18 pm

The integers x and y are both positive, the remainder when x is divided by 12 is 7, and the remainder when y is divided by 12 is 3. Each of the following is a possible value of 2x+y EXCEPT

A. 125
B. 101
C. 77
D. 51
E. 41

Answer: D

Solution:

We see that x = 12m + 7 and y = 12n + 3 where m and n are non-negative integers. Therefore,

2x + y = 2(12m + 7) + 12n + 3 = 24m + 14 + 12n + 3 = 24m + 12n + 17 = 12(2m + n + 1) + 5

We see that 2x + y is 5 more than a multiple of 12 and every number in the given choices is 5 more than a multiple of 12 except 51 (notice that 51 is 3 more than 48). Therefore, 51 is the correct answer.

Answer: D