# Practice GMAT Problem Solving Question

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What is the units digit of (2)

^{3}(3)^{3}(4)^{3}(5)^{7}(6)^{2}(7)^{2}?Correct Answer:

**A**- At first, this question looks daunting. Do we have to determine what all of these numbers are, raised to their respective powers, and then multiply them by one another? Recognize that if we just figure out what the units (ones) digit is for all of these numbers, we then just have to multiply the units digits by one another, which is much more manageable. This is because, to obtain the units digit of a product of two numbers, we only need to multiply the units digit of these two numbers by each other. Then the units digit of that product will be the units digit of the whole product. For example, the units digit of 38 * 9317 = 6, because 8 * 7 = 56, and the units digit of 56 is 6.
- Thus, let us first determine the units digit of the numbers in this expression.
- (2)
^{3}= 2 * 2 * 2 = 8

So the units digit of (2)^{3}is 8 - (3)
^{3}= 3 * 3 * 3

= 9 * 3

= 27

So the units digit of (3)^{3}= 7 - (4)
^{3}= 4 * 4 * 4

= 16 * 4

= 64

The units digit of (4)^{3}= 4 - For (5)
^{7}, recognize that 5 to any power greater than 0 must have a 5 in the units digit. For example, 5 * 5 = 25; 25 * 5 = 125, etc.

Therefore the units digit of (5)^{7}= 5 - At this point, we can stop, if we recognize that 5 (i.e., the units digit of (5)
^{7}) * 4 (i.e., the units digit of (4)^{3}) = 20. Remember that after determining all of the units digits, we were going to multiply them by one another to determine the units digit of the whole product. The fact that 5 * 4 = 20 is significant because, taking 0 as the units digits of this product, we then will multiply 0 by all of the other units digits we determine. This product, however, no matter what the other units digits are, must be 0, because any number multiplied by 0 is 0. Therefore, our units digit must be 0, answer choice (A).

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