Problem Solving

[GMAT math practice question]

If a positive integer t is not divisible by 5, how many possible different remainders can t^4 have when it is divided by 5?

A. one
B. two
C. three
D. four
E. five

=>

If t has remainder 1 when it is divided by 5, t^4=~1^4 has remainder 1 when it is divided by 5.
If t has a remainder 2 when it is divided by 5, t^4=~2^4=~16 has remainder 1 when it is divided by 5.
If t has remainder 3 when it is divided by 5, t^4=~3^4=~81 has remainder 1 when it is divided by 5.
If t has remainder 4 when it is divided by 5, t^4=~4^4=~256 has remainder 1 when it is divided by 5.

t^4 has the unique remainder, which is 1, for all values of t.

Therefore, the answer is A.
Answer: A

If 't' is not divisible by 5, remainder will be either 1,2,3 or 4.
Checking for all possible remainders, we can replace 't' in 't^4' with all possible remainder.
$$If\ t=1,\ then\ t^4=1^4=1\ as\ the\ remainder.$$
$$If\ t=2,\ then\ t^4=2^4=16\ as\ the\ remainder.$$
But, 16= (3*5) + 1. So, 1 is the remainder
$$If\ t=3,\ then\ t^4=3^4=81\ as\ the\ remainder\ but\ \left(16\cdot5\right)+1=81.$$
So, the remainder is 1.
$$If\ t=4,\ then\ t^4=4^4=256\ as\ the\ remainder\ but\ \left(51\cdot5\right)+1=256$$
The remainder is also 1.

Therefore, in all case, 1 (one) is the remainder.

Hence, option A is correct