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Number Systems

by sukhman » Thu Oct 17, 2013 2:26 am
If x is a positive integer, what is the units digit of (24)^(2x + 1)(33)^(x + 1)(17)^(x + 2)(9)^(2x)?
(A) 4 (B) 6 (C) 7 (D) 8 (E) 9
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by GMATGuruNY » Thu Oct 17, 2013 3:06 am
sukhman wrote:If x is a positive integer, what is the units digit of (24)^(2x + 1)(33)^(x + 1)(17)^(x + 2)(9)^(2x)?
(A) 4 (B) 6 (C) 7 (D) 8 (E) 9
To determine the units digit of PRODUCT XYZ:
1. Multiply (units digit of x)(units digit of y)(units digit of z).
2. The units digit of the resulting product = the units digit of product xyz.

Let x=1.

Since 4^(2x+1) = 4^(2*1 + 1) = 4³ = 64, the units digit of 24^(2x+1) = 4.
Since 3^(x+1) = 3^(1 + 1) = 3² = 9, the units digit of 33^(x+1) = 9.
Since 7^(x+2) = 7^(1 + 2) = 7³ = 343, the units digit of 17^(x+2) = 3.
Since 9^(2x) = 9^(2*1) = 9² = 81, the units digit of 9^(2x) = 1.

Multiplying the resulting units digits, we get:
4*9*3*1 = 108.
Thus, the units digit of (24)^(2x + 1)(33)^(x + 1)(17)^(x + 2)(9)^(2x) = 8.

The correct answer is D.
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by mevicks » Thu Oct 17, 2013 8:18 am
sukhman wrote:If x is a positive integer, what is the units digit of (24)^(2x + 1)(33)^(x + 1)(17)^(x + 2)(9)^(2x)?
(A) 4 (B) 6 (C) 7 (D) 8 (E) 9
Image

Consider the only the units digits of the equation & use the cyclicity property to quickly find out the units digit of the final result.

Image

Check using two values
Let x = 1
Image
By cyclicity --> 4*9*3*1
Units digit = 8

Let x = 2
Image
By cyclicity --> 4*7*1*1
Units digit = 8

Answer D
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