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If a is an integer, what is the units digit of a^18?

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If a is an integer, what is the units digit of a^18?

by Gmat_mission » Mon Jun 10, 2019 5:28 am

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If $$a$$ is an integer, what is the units digit of $$a^{18}?$$

(1) $$a^2$$ has a units digit of $$9$$
(2) $$a^7$$ has a units digit of $$3$$

[spoiler]OA=D[/spoiler]

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by ceilidh.erickson » Tue Jun 11, 2019 12:09 pm

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When dealing with UNITS DIGITS and exponents, we must look for a pattern.

If $$a$$ is an integer, what is the units digit of $$a^{18}?$$

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

We can express $$a^{18}$$ as $$(a^2)^9$$ . If $$a^2$$ has a units digit of $$9$$, then $$a^{18}$$ will have the same units digit as $$9^9$$.

We don't even have to calculate - we know that $$9^9$$ will have a knowable units digit. Just as a reminder, though, units digits will always follow a pattern. Here is the pattern for the units digit of 9:
$$9^1$$ = 9
$$9^2$$ = ...1
$$9^3$$ = ...9
$$9^4$$ = ...1

9 raised to any odd exponent will end in 9, and to any even exponent will end in 1. Thus, $$9^9$$ will end in 9.

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

We cannot easily relate $$a^7$$ to $$a^{18}$$. So, we must look for numbers that have a units digit of 3 when raised to the 7th power. You might already know that 3 and 7 are the only units digits that can yield a 3. (No even digits when raised to a power will ever end in an odd digit, and $$1^x$$ and $$5^x$$ will always end in 1 and 5, respectively. As shown above, $$9^x$$ can only end in 9 or 1).

Test the powers of 3:
$$3^1$$ = 3
$$3^2$$ = 9
$$3^3$$ = ...7
$$3^4$$ = ...1
$$3^5$$ = ...3
$$3^6$$ = ...9
$$3^7$$ = ...7
This doesn't fit.

Try 7:
$$7^1$$ = 7
$$7^2$$ = ...9
$$7^3$$ = ...3
$$7^4$$ = ...1
$$7^5$$ = ...7
$$7^6$$ = ...9
$$7^7$$ = ...3

This fits! We know that the units digit of $$a$$ must be 7. This is sufficient to know what the units digit of $$a^{18}$$ would be.

Again, don't waste your time on test day figuring out what the units digit actually is. But for reference, if $$a^{18}$$ is $$...7^{18}$$, we can use the fact 7's repeat every 4 powers (see above: $$7^1$$ has the same units digit as $$7^5$$.) Every $$7^{4x}$$ will end in a 1. Thus,
$$7^{16}$$ = ...1
$$7^{17}$$ = ...7
$$7^{18}$$ = ...9

Ceilidh Erickson
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EdM in Mind, Brain, and Education

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by ceilidh.erickson » Tue Jun 11, 2019 12:16 pm

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Ceilidh Erickson
Manhattan Prep GMAT & GRE instructor
EdM in Mind, Brain, and Education