BTGModeratorVI wrote: ↑

Tue Feb 16, 2021 8:13 am

If n = (33)^43 + (43)^33 what is the units digit of n?

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

Answer: A
Source: official guide

Look for a pattern

33^1 = 3
33^2 = (33)(33) = ---9 [aside: we need not determine the other digits. All we care about is the units digit]
33^3 = (33)(33^2) = (33)(---9) = ----7
33^4 = (33)(33^3) = (33)(---7) = ----1
33^5 = (33)(33^4) = (33)(---1) = ----3

NOTICE that we're back to where we started.
33^5 has units digit 3, and 33^1 has units digit 3
So, at this point, our pattern of units digits keep repeating 3, 9, 7, 1, 3, 9, 7, 1, 3,...
We say that we have a "cycle" of 4, which means the digits repeat every 4 powers.

Now that we know the pattern has a cycle of 4, let's examine powers where the exponent is a multiple of 4
We know that 33^4 has units digit 1
So this means 33^8 has units digit 1
And 33^12 has units digit 1
And 33^16 has units digit 1
And so on

To find the unit's digit of (33)^43, let's find the nearest multiple of 4 that's less than 43.
40 is a multiple of 4, which means 33^40 has units digit 1
From here we can continue our pattern to see that
33^41 has units digit 3
33^42 has units digit 9
33^43 has units digit 7
--------------------------------------

Since 33 and 43 have the same units digit, 33 and 43 will have the exact same cycle of 4
So, to find the unit's digit of (43)^33, we'll first find the nearest multiple of 4 that's less than 33.
32 is a multiple of 4, which means 43^32 has units digit 1
From here we can continue our pattern to see that
43^33 has units digit 3
---------------------------------------

So, (33)^43 + (43)^33 = ----7 + ----3 = ----0

Answer: A

Here's an article I wrote on this topic (with additional practice questions): https://www.gmatprepnow.com/articles/un ... big-powers

Cheers,
Brent