What is the units digit of a18 ? a2 has a units d

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is below explanation from VERITAS is correct - plz check

What is the units digit of
a18
?

a2
has a units digit of 9
a7
has a units digit of 3

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient

EACH statement ALONE is sufficient to answer the question asked

Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
solution

D. While statement 1 allows for a to have a units digit of either 3 or 9, in either case the units digit of
a18
will be 9. For any units digit patterns, the fourth power is the "anchor" - whatever the units digit of the fourth power is will be the same for every exponent that is a multiple of four. Knowing that
a2
has a units digit of 9 tells us that for any exponent in the form 4x + 2 the units digit will also be 9.

Statement 2 is also sufficient. No number with an even units digit can ever be raised to any integer power and have a units digit of 3. A number with a units digit of 1 or of 5 will never change its units digit when raised to any integer power, and a number will a units digit of 9 can only have a units digit of 9 or 1 when raised to any integer power. This means that if
a7
has a units digit of 3, the only numbers that MIGHT work are a ends in 3 and a ends in 7. If we try a ends in 3, the units digit pattern goes
a1
= ends in 3,
a2
= ends in 9,
a3
= ends in 7,
a4
= ends in 1, and repeats in blocks of four, so
37
must end in 7, meaning a number ending in 3 is also not a possible value for a. a must thus end in a 7, so we can compute the units digit of a number ending in 7 to the 18th power;

The correct answer is D.

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by [email protected] » Mon Jan 14, 2013 12:49 pm
Yep, it's good.
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by ceilidh.erickson » Tue Jan 15, 2013 7:51 am
Whenever you see a units digit (UD) question involving exponents, the first thing to do is to establish the pattern of the UDs. For example, the pattern of UDs with powers of 3 is as follows:
3^1 = 3
3^2 = 9
3^3 = 7 (only worry about the UD)
3^4 = 1
3^5 = 3
3^6 = 9

You can see that the UD of powers of 3 repeat every 4 exponents. All UDs have similar patterns.

If the question is asking "what is the UD of a^18?", we either need to establish what "a" is, or some smaller power of "a" that we can take to the 18th.

(1) a^2 has a units digit of 9

This statement is not enough to determine the UD of a - it could be 3 or 7. 3^2 = 9, and 7^2 = 49, so either of those would fit. But remember - the question isn't asking "what is the UD of a?" It's asking "what is the UD of a^18?" We know that a^18 = (a^2)^9. So what happens when we take something with a UD of 9^9?
9^1 = 9
9^2 = 1 (UD only)
9^3 = 9
9^4 = 1
So for any odd power of 9, the UD is 9. It doesn't matter whether the UD of "a" is 3 or 7, because when we raise it to the 9th power, the UD will be 9.

(2) a^7 has a units digit of 3

This statement tells us that the UD of "a" must be 7, since only 7's will have a UD of 3 when taken to the 7th power. If we know the UD of "a", we definitely know the UD of a^18.

For more on units digits, see these posts:
https://www.beatthegmat.com/what-is-the- ... tml#544267 (this is a very similar problem)
https://www.beatthegmat.com/if-r-s-and-t ... tml#548713
https://www.beatthegmat.com/if-n-and-m-a ... tml#544266
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education