BTGmoderatorDC 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

OA B

Source: Manhattan Prep

Given: 2^a*8^b*4^(a+b)

=> [2^a]*[(2^3)^b]*[2^{2^(a+b)}] = 2^a*2^(3b)*2^{2(a+b)} = 2^(a + 3b + 2(a+b)) = 2^{3(a + b)}

So, we have to find out the units digit of 2^{3(a + b)}.

Let's see the power cycle of 2.

a. 2^1 = 2 => Units digit = 2;
b. 2^2 = 4 => Units digit = 4;
c. 2^3 = 8 => Units digit = 8;
d. 2^4 = 16 => Units digit = 6;

e. 2^5 = 32 => Units digit = 2 = Same as that for 2^1;
f. 2^6 = 64 => Units digit = 4 = Same as that for 2^2;
g. 2^7 = 128 => Units digit = 8 = Same as that for 2^3;
h. 2^8 = 256 => Units digit = 6 = Same as that for 2^4;

You'll notice that after every 4th exponent, the units digit repeats.

Thus,

a. For 2^(4n + 1), Units digit = 2;
b. For 2^(4n + 2), Units digit = 4;
c. For 2^(4n + 3), Units digit = 8;
d. For 2^(4n + 4), Units digit = 6

So, to get the units digit of 2^{3(a + b)}, we must know whether the units digit of 3(a + b) is divisible by 4, if not what remainder it leaves when divided by 4.

Question rephrased: What's the remainder when 3(a + b) is divided by 4?

Let's take each statement one by one.

(1) b = 24 and a < 24

Case 1: Say a = 4, then 3(a + b) = 3(4 + 24) = 28*3, it is divisible by 4. Remiander is 6.
Case 1: Say a = 1, then 3(a + b) = 3(1 + 24) = 75, it is NOT divisible by 4. Remiander is 3.

No unique answer. Insufficient.

(2) The greatest common factor of a and b is 12.

=> a = 12x and b = 12y, where x and y are co-prime

Thus, 3(a + b) = 3(12x + 12y) = 36(x + y), it is divisible by 4. Remiander is 6.

Sufficient.

The correct answer: B

Hope this helps!

-Jay
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