Q55. GMAT Math DS - Powers and units digit

Q. If a is an integer. What is the units digit of a^36?
a) a^2 has 9 as the units digit
b) a^3 has 3 has the units digit

PLEASE PROVIDE YOUR REASONING AND NOT JUST THE ANSWER CHOICES.

Difficulty Level : Moderate
Time to Solve : <70 secs

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B
As Statement 2 is sufficient to get the number as 7
Time 90 Secs

IMO D

Statemenet 1 gives units place as 1

Statement 2 gives units place as 1

D .. guessing a number 7, 7*7=49 and 7*7*7=343. Statement 1 and 2 both sufficient separately ..

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Nurul Hai

statement 1 can be true for 3 & 7

statement 2 can be true for 7 only

Hence B

1) a^36=(a^2)^18=(...9)^18, from this we can find the units digit
2) a^36=(a^3)^12=(...3)^12, from here we can find the units digit.
As mentioned by Mickey, both times the units digit equals to 1

D – 90 secs

From a: Since a^2 has 9 as the units digit, “a” has either 3 or 7 as the ones digit.

· Number 3: The number 3 has a repeating series of the ones digit every 4th power. e.g. 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 à 1 is the one digit of this number (= 7*3), 3^5 à 3 is the ones digit of this number (= 1*3). Since there is a repeating pattern every 4th power, the one digit of a^36 can be deduced to be 1 since 36 is divisible by 4.

· Number 7: The number 7 also has repeating series of the ones digit every 4th power. e.g. 7^1 = 7, 7^2 = 49, 7^3 à 3 is the ones digit of this number (=9*7), 7^4 à 1 is the ones digit of this number (=3*7), 7^5 à 7 is the ones digit of this number (=1*7). Since there is a repeating pattern every 4th power, the one digit of a^36 can be deduced to be 1 since 36 is divisible by 4.

Either number 3 or 7 will produce 1 as the ones digit of a^36th. Sufficient.

From b: Since a^3 has a 3 in the units digit, “a” has to have 7 as the ones digit. Since we know the ones digit of a^36 is equal to 1 from the analysis shown above, this is Sufficient also.

Hence D. à Either statement by itself is sufficient to answer the question.

D............
agree with rkandell

Probably, if it were indicated that "a" is integer, the conditions would be a little bit more correct. Whatever, even if it's not integer (let's say (19)^0,5 in the first case and 3^(1/3) in the second), the answer is D. Both will end with 1

Clerk,
Good point about "a" being an integer. I didn't think about the possibilities of "a" not being an integer.

Still no OA and OE

Good point clerk, a should have been given as an integer ...

What is the units digit of a^36?
a) a^2 has 9 as the units digit
b) a^3 has 3 has the units digit

OA: D

a) a^2 has 9 => (a^2)^18 will have 9^18 ... which gives 1 as units digit. suff

b) a^3 has 3 => (a^3)^12 will have 3^13 .. which gives 1 as units digit. suff

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D