Problem Solving

Vincen wrote:What is the units digit of 248^20?

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

Look for a pattern

248^1 = 248
248^2 = (248)(248) = ---4 [aside: we need not determine the other digits. All we care about is the units digit]
248^3 = (248)(248^2) = (248)(---4) = ----2
248^4 = (248)(248^3) = (248)(---2) = ----6
248^5 = (248)(248^4) = (248)(---6) = ----8

NOTICE that we're back to where we started.
248^5 has units digit 8, and 248^1 has units digit 8
So, at this point, our pattern of units digits keep repeating 8, 4, 2, 6, 8, 4, 2, 6, 8,...
We say that we have a "cycle" of 4, which means the digits repeat every 4 powers.

So, we get:
248^1 = --8
248^2 = ---4
248^3 = ----2
248^4 = ----6
248^5 = ----8
248^6 = ---4
248^7 = ----2
248^8 = ----6
248^9 = ----8
248^10 = ----4
etc.

Notice that when the exponent is a MULTIPLE of 4 (4, 8, 12, 16, ...), the units digit will be 6
Since 20 is a MULTIPLE of 4, we know that the units digit of 248^20 will be 6
Answer: D

Here's an article I wrote on this topic (with additional practice questions): https://www.gmatprepnow.com/articles/un ... big-powers

Cheers,
Brent

mbouayad14 wrote:20 is a multiple of 2....so why couldnt rhe answer be 4???

That would be true if it were the case that ALL exponents that are multiples of 2 yielded units digit 4

Cheers,
Brent

Vincen wrote:What is the units digit of 248^20?

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

Since we only care about units digits, we can rewrite the expression as:

8^20

Let's start by evaluating 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 an 8 as the units digit. Thus:

8^20 has a units digit of 6.

Answer: D

Vincen wrote: How can I find the units digit without making the whole calculus?

You should never try to calculate the actual value when asked for a units digit! It's all about finding the PATTERNS in the units digits. Try to find the patterns in these examples:
https://www.beatthegmat.com/if-r-s-and-t ... tml#548713
https://www.beatthegmat.com/if-n-and-m-a ... tml#544266
https://www.beatthegmat.com/what-is-the- ... tml#554073
https://www.beatthegmat.com/what-is-the- ... tml#544267
https://www.beatthegmat.com/what-is-the- ... tml#554073