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What is the units digit of the product (32^2

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by Brent@GMATPrepNow » Sun Apr 09, 2017 6:51 am
What is the units digit of the product (32^28) (33^47) (37^19)?

A) 0
B) 2
C) 4
D) 6
E) 8

Answer: D

Source: www.gmatprepnow.com
Difficulty level: 600 - 700

Hint #1: [spoiler]There's a fast approach and a slow approach[/spoiler]

Hint #2: Notice that the largest exponent is equal to the sum of the other two exponents
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GMAT/MBA Expert

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by Brent@GMATPrepNow » Mon Apr 10, 2017 7:56 am
Brent@GMATPrepNow wrote:What is the units digit of the product (32^28) (33^47) (37^19)?

A) 0
B) 2
C) 4
D) 6
E) 8
-----ASIDE-----------------------------------------
There are some "nice" numbers that, when raised to various powers, ALWAYS have the same units digit.
For example, the units digit of 70^n will be 0 FOR ALL POSITIVE INTEGER VALUES OF N
Likewise, the units digit of 91^n will be 1 FOR ALL POSITIVE INTEGER VALUES OF N
And the units digit of 86^n will be 6 FOR ALL POSITIVE INTEGER VALUES OF N
-----NOW ONTO THE QUESTION-----------------

Notice that the exponent 47 is equal to the SUM of the other two exponents (28 and 19)
So, it might be useful to take 33^47 and REWRITE it as (33^28)(33^19)
NOTE: later on, we'll apply a nice exponent rule that says "(a^n)(b^n) = (ab)^n"

We get: (32^28)(33^47)(37^19) = (32^28)(33^28)(33^19)(37^19)
= (32^28 x 33^28)(33^19 x 37^19)
= (32 x 33)^28 (33 x 37)^19 [applied above rule]
=(---6)^28 (---1)^19 [I'm focusing solely on the units of each product. So, I use "---" to represent the other digits]
=(----6)(----1) [When ----6 is raised to any power the units digit is always 6. The same applies to ----1]
= -------6

Answer: D

Cheers,
Brent
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