Problem Solving

Vincen wrote:A number when divided by 105 leaves 99 as remainder. What will be the remainder if the number is divided by 21?

A) 9
B) 14
C) 20
D) 6
E) 15

A lot of integer properties questions can be solved by plugging in numbers.

In this case, we'll use a nice 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......
A number when divided by 105 leaves 99 as remainder
Possible values of the number: 99, 99+105, 99+(2)(105), 99+(3)(105), 99+(4)(105), etc
We can use ANY of the above values to answer the question.
So, let's use the smallest value (99)

What will be the remainder if the number is divided by 21?
99 divided by 21 equals 4 with remainder 15
Answer: E

Notice that we get the same result if we test other possible values.
For example, let's test the next possible value: 99+105
99+105 = 204
204 divided by 21 equals 9 with remainder 15
Answer: still E

RELATED VIDEO: https://www.youtube.com/watch?v=XCxgIR042tE

Cheers,
Brent

Vincen wrote:A number when divided by 105 leaves 99 as remainder. What will be the remainder if the number is divided by 21?

A) 9
B) 14
C) 20
D) 6
E) 15

We can let n = the number and we have:

n = 105Q + 99

Thus, when that number is divided by 21, we have:

(105Q + 99)/21

5Q + 99/21

Since 5Q is an integer, the remainder will come from 99 divided by 21:

99/21 = 4 R 15

We see that the remainder is 15.

Answer: E