12 Days of Christmas GMAT Competition - Day 2: What is the last digit

Question

12 Days of Christmas GMAT Competition with Lots of Fun

What is the last digit in the expanded value of  2^{394} – 6^{88} ?

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

Bunuel · Accepted Answer

Expert's Global Official Video Explanation

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DHRJ0032 · Answer

power of 2 will end with 4 and power of 6 will end with 6
so , 8 will be the answer.

D it is

gmatophobia · Answer

We can consider the cyclicity of each number to be 4

2 ^ {394}   -

\frac{394 }{ 4} , the remainder is 2 hence the last digit is 4

6 ^ {88}   -

\frac{88 }{ 4} , the remainder is 8 hence the last digit is 6

So the last digit in the expanded value will be 4 - 6 , which is 8

IMO D

Lizaza · Answer

Let's look at it the following way: no matter how high the power, we can find the last digit of the number by looking at the cycle, in which last digits change one ahother with power growth.

Starting with 2:

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = _6
***
2^5 = _2
...

What we see, is that after 4 steps, the cycle begins again: 2 - 4 - 8 - 6, 2 - 4 - 8 - 6, and so on. Therefore, we only need to calculate how many 'fours' are inside 394, to find the last digit.
394 is not divisible by 4, but 392 is. Therefore, the full cycle ends, and we have two more steps that is, 2 and 4. Right, the last digits of the first number is 4!

As for 6, it's even simpler: no matter the power, the ending is always 6. Let's check: 6*6 = 36, 6*6*6 = 216, and so on.

Therefore, whatever the actual numbers, the answer is _________4 - ________6 = ______8
It's the option  D) 8.

MGMAT780 · Answer

Hello Bunuel

Can you please let us know why do we need to add 10 to get the last digit

Bunuel · Answer

The units digit of 2^394 is 4, and the units digit of 6^88 is 6. Since the first number (2^394), is larger than the second one (6^88), we subtract the smaller number with the units digit of 6 from the larger number with the units digit of 4, which gives a result with the units digit of 8. For example, 14 - 6 = 8.

Hope it's clear.

bumpbot · Answer

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12 Days of Christmas GMAT Competition - Day 2: What is the last digit

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12 Days of Christmas GMAT Competition - Day 2: What is the last digit

Prep Toolkit

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My Rewards