Question

Each of the blank spaces below represents a single digit non-negative integer. Which of the following could be the square of a positive integer y? A 33_,_42 B 45_,_88 © 57_,_36 D 80_,_03 E 82_,_60​

Answer 1

Step-by-step explanation:

If n = (33)^43 + (43)^33 what is the units digit of n?

A. 0

B. 2

C. 4

D. 6

E. 8

First of all, the units digit of (33)^43 is the same as that of 3^43 and the units digit of (43)^33 is the same as that of 3^33. So, we need to find the units digit of 3^43 + 3^33.

Next, the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}:

3^1=3 (the units digit is 3)

3^2=9 (the units digit is 9)

3^3=27 (the units digit is 7)

3^4=81 (the units digit is 1)

3^5=243 (the units digit is 3 again!)

...

Thus:

The units digit of 3^43 is the same as the units digit of 3^3, so 7 (43 divided by the cyclicity of 4 gives the remainder of 3).

The units digit of 3^33 is the same as the units digit of 3^1, so 3 (33 divided by the cyclicity of 4 gives the remainder of 1).

Therefore the units digit of (33)^43 + (43)^33 is 7 + 3 = 0.

Answer: A.