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
