If \(23^3\cdot 19^4\cdot14^2 = x\), what is the unit's digit

Gmat_mission wrote:If \(23^3\cdot 19^4\cdot14^2 = x\), what is the unit's digit of \(x\)?

A. 2
B. 3
C. 6
D. 7
E. 9

[spoiler]OA=A[/spoiler]

Source: Princeton Review

The units digit of 23^3 is 7, of 19^4 is 1, and of 14^2 is 6. So the units digit of the given expression is the same as the units digit of 7 x 1 x 6 = 42, which is 2.

Answer: A

Hi All,

We're told that X = (23^3)(19^4)(14^2). We're asked for the unit's digit (re: the "ones digit") of X. While this question might look a little 'scary' it involves some Number Property rules that you can take advantage of and you really just need to do some basic multiplication to answer the question.

To start, 'squaring' an EVEN number will get us an EVEN number. So 14^2 is EVEN. Next, any time you multiply an integer by an EVEN number, the product will also be EVEN. Here, we're multiplying 3 integers together - and we already know that 14^2 is EVEN, so the value of X will also be EVEN. This means that the units digit of X must be an EVEN number (0, 2, 4, 6 or 8). At this point, we can eliminate Answers B, D and E.

To find the units digit of a number, we just have to multiply the units digits of the products together...

14^2 = (14)(14).... here, the product of the units digits is (4)(4)..... 16..... the units digit of the end product is 6
23^3 = (23)(23)(23).... here, the product of the units digits is (3)(3)(3)..... 27..... the units digit of the end product is 7
19^4 = (19)(19)(19)(19).... here, the product of the units digits is (9)(9)(9)(9)..... 6561..... the units digit of the end product is 1

Multiplying those three units digits together, we get (6)(7)(1) = 42... so the units digit of X is 2.

Final Answer: A

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