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If a and b are distinct positive integers

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If a and b are distinct positive integers

by PGMAT » Thu Jul 05, 2012 7:12 pm
If a and b are distinct positive integers, what is the units digit of 2^a 8^b 4^(a+b)?
1. b=24 and a<24
2. The greatest common factor of a and b is 12

[spoiler]OA is B. Can some one please explain how to solve this? I took different values for a and b with GCF 12 for statement 2 and got different units digit[/spoiler]

Thanks.
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Source: — Data Sufficiency |

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by Anurag@Gurome » Thu Jul 05, 2012 7:35 pm
PGMAT wrote:If a and b are distinct positive integers, what is the units digit of 2^a 8^b 4^(a+b)?
1. b=24 and a<24
2. The greatest common factor of a and b is 12
8^b = (2^3)^b = 2^[3b]
4^(a + b) = (2^2)^(a + b) = 2^[2(a + b)]

(2^a)*(8^b)*(4^(a + b)) = (2^a)*(2^[3b])*(2^[2(a + b)]) = 2^(a + 3b + 2(a + b)) = 2^(3a + 5b)

Now, the units digit of powers of 2 (for positive integral powers) follow the following pattern
  • Unit's digit of 2^(multiple of 4) = 6
    Unit's digit of 2^(multiple of 4 + 1) = 2
    Unit's digit of 2^(multiple of 4 + 2) = 4
    Unit's digit of 2^(multiple of 4 + 3) = 8
Statement 1: As we don't know the definite value or nature of a, we cannot determine the unit's digit of the given expression.

Not sufficient

Statement 2: As GCF of a and b is 12, a and b both must be multiple of 12. Hence, both of them are multiple of 4.
Therefore, (3a + 5b) is also multiple of 4.
Hence, uni'ts digit of the given expression is 6

Sufficient

The correct answer is B.
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