If r, s, and t are all positive integers, what is the remain

[BTGmoderatorDC]

If r, s, and t are all positive integers, what is the remainder of 2^p/10, if p = rst?

(1) s is even
(2) p = 4t

OA B

Source: Manhattan GMAT

[Jay@ManhattanReview]

If r, s, and t are all positive integers, what is the remainder of 2^p/10, if p = rst?

(1) s is even
(2) p = 4t

OA B

Source: Manhattan GMAT

Remainder rule for division by 10:

Note that when a positive integer is divided by 10, the remainder is its units digit. For example, 50 divided by 10 returns a remainder of 0; 65 divided by 10 returns a remainder of 5; and 87 divided by 10 returns a remainder of 7.

Power cycle of 2:

The units digit of 2 with exponents of a positive integer repeats in a cycle of 4:

The units digit of 2^1 (= 2) = 2;
The units digit of 2^2 (= 4) = 4;
The units digit of 2^3 (= 8) = 8;
The units digit of 2^4 (= 16) = 6;

The units digit of 2^5 (= 32) = 2
.
.
.
And so on...

Let's take each statement one by one.

(1) Since s is even, p is even. This implies that the units digit of 2^p is either 4 or 6 (see the power cycle given above). No unique value of the remainder. Not sufficient.

(2) p = 4t implies that p is a multiple of 4; thus, the units digit of 2^p or 2^(4t) is the same as the units digit of 2^4, which is 6 (see the power cycle given above). Thus, the remainder is 6. Sufficient.

The correct answer: B

Hope this helps!

-Jay
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