When integer a is divided by 4 the remainder is 2, and when integer b is divided by 5 the remainder is 1. How many integ

BTGModeratorVI wrote: ↑

Wed Dec 16, 2020 12:33 pm

When integer a is divided by 4 the remainder is 2, and when integer b is divided by 5 the remainder is 1. How many integers between 20 and 29, inclusive, CANNOT be the sum a + b?

A. Zero
B. One
C. Two
D. Three
E. Four

Answer: A
Source: Veritas Prep

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.
------NOW ONTO THE QUESTION----------------------------

When integer a is divided by 4 the remainder is 2
So, the possible values of a are: 2, 6, 10, 14, 18, 22, 26,....

When integer b is divided by 5 the remainder is 1
So, the possible values of b are: 1, 6, 11, 16, 21, 26, 31,....

How many integers between 20 and 29, inclusive, CANNOT be the sum a + b?
Let's see which sums we CAN get:
20 = 14 + 6
21 = 10 + 11
22 = 6 + 16
23 = 2 + 21
24 = 18 + 6
25 = 14 + 11
26 = 10 + 16
27 = 26 + 1
28 = 22 + 6
29 = 18 + 11
We can get ALL of the sums.

Answer: A