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

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

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

Since we only care about units digits, the units digit of 18^47 will be the same as the units digit of 8^47.

Let's determine the units digit of 8^47 by determining the pattern of the units digits of 8^n for positive integer values of n. That is, let's look at the pattern of the units digits of powers of 8. When writing out the pattern, notice that we are ONLY concerned with the units digit of 8 raised to each power.

8^1 = 8

8^2 = 4

8^3 = 2

8^4 = 6

8^5 = 8

The pattern of the units digit of powers of 8 repeats every 4 exponents. The pattern is 8-4-2-6. In this pattern, all positive exponents that are multiples of 4 will produce a 6 as their units digits. Thus:

8^48 has a units digit of 6.

8^47 has a units digit of 2.

Answer: B