Problem Solving

What is the units digit of 18^47 ?

A. 0
B. 2
C. 4
D. 6
E. 8

How can I solve this PS question? It is a large list. Experts, may you give a fast way?

M7MBA wrote:What is the units digit of 18^47 ?

A. 0
B. 2
C. 4
D. 6
E. 8

How can I solve this PS question? It is a large list. Experts, may you give a fast way?

When we're dealing with a units digit question, we need not worry about anything other than the units digit. In other words, the units digit of 18^47 is the same as the units digit of 8^47.

8^47 = (2^3)^47 = 2^141

Now all we have to do is establish the units digit pattern for base 2.

2^1--> units of 2
2^2 --> units of 4
2^3 --> units of 8
2^4 --> units of 6

And then the pattern resets,

2^5--> units of 2
2^6 --> units of 4
2^7 --> units of 8
2^8 --> units of 6

The upshot is that the pattern repeats after four terms. Put another way, if the exponent is a multiple of 4, we know there will be a units digit of 6. 2^8 will have a units digit of 6, and 2^12 will have a units digit of 6, etc.

The closest multiple of 4 to 141 is 140. If 2^140 gives a units digit of 6, then 2^141 will yield the next number in the pattern, giving a units of 2. So the answer is B