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If a is an integer, what is the units digit of a^18? (1) a^

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If a is an integer, what is the units digit of a^18? (1) a^

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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

OA D

Source: Veritas Prep

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BTGmoderatorDC wrote:
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

OA D

Source: Veritas Prep
If we want to know the units digit of a^18, we must know something about the units digit of a.

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

Test digits to see if there is more than one units digit that will yield a 9 when squared:
a = 1 --> a² = 1 --> no.
a = 2 --> a² = 4 --> no.
a = 3 --> a² = 9 --> YES.
a = 4 --> a² = 16 --> no.
a = 5 --> a² = 25 --> no.
a = 6 --> a² = 36 --> no.
a = 7 --> a² = 49 --> YES
a = 8 --> a² = 64 --> no.
a = 9 --> a² = 81 --> no.

Thus, 'a' can have a units digit of 3 or 7. For each of these, though, a^18 = (a^2)^9 . So we'd be taking a number with a units digit of 9 and taking it to the 9th power. This will always yield a units digit of 9. (Odd powers of 9 always end in a 9). Sufficient.

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

Units digit patterns always repeat after every 4th power. Eg:
x = 2
x^2 = 4
x^3 = 8
x^4 = 16
x^5 = 32
x^6 = 64
etc.

Some digits (0, 1, 5, and 6) always stay constant, and some (4 and 9) repeat every 2 powers, but every digit repeats at least every 4th power. So, every a^(4n + 3) will have the same units digit. Or more simply put, a^7 will have the same units digit as a^3. Let's just test a^3 for every digit (ignoring everything except units digits):

a = 1 --> a^3 = 1 --> no.
a = 2 --> a^3 = 8 --> no.
a = 3 --> a^3 = 7 --> no.
a = 4 --> a^3 = 4 --> no.
a = 5 --> a^3 = 5 --> no.
a = 6 --> a^3 = 6 --> no.
a = 7 --> a^3 = 3 --> YES.
a = 8 --> a^3 = 2 --> no
a = 9 --> a^3 = 9 --> no.

The units digit of 'a' must be 7. This is sufficient to determine the units digit of a^18.

The answer is D.

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Manhattan Prep GMAT & GRE instructor
EdM in Mind, Brain, and Education
Harvard Graduate School of Education


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