Source: Official Guide for the GMAT 13th Ed. Problem Solving; #163 Official Guide for the GMAT 2015 14th Ed. Problem Solving; #163

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# If n = (33) + (43), what

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

### 2 Explanations

1

ANNA CHENGUELLY

Hello,
Could you please kindly help to understand why do we convert exponent value in to multiple of 4? Thank you

Apr 24, 2017 • Comment

Cydney Seigerman, Magoosh Tutor

Happy to help :)

It turns out that for exponents of a given base, the units digit follows a repeating pattern. When we determine this pattern, we can use it to find out where the pattern will be at the desired power.

In this practice problem, the units digit of both numbers is 3. So, we will want to look at the exponents of 3 to find the pattern that the units digit will follow:

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243

We can see that there are 4 different values the units digits can be when the base number ends in 3:

3, 9, 7, or 1

So, we can consider groups of 4 to help us figure out at which point of the pattern we will be given the exponents in the problem:

33^43 --> we can repeat the pattern 3, 9, 7, 1 a total of 10 times. In the 11th cycle of the pattern, we have

33^41 --> ends in 3
33^42 --> ends in 9
33^43 --> ends in 1

43^33 also has a base (43) that ends in 3. So, we will use the same pattern of 4. In this case, we can repeat the pattern 3, 9, 7, 1 a total of 8 times. In the 9th cycle, we have

43^33 --> end in 3

In both cases, the 4 comes from the number of different possible units digits for exponents of a number than ends in 3.

I hope this clears up your doubts :)

Michael Onyemelukwe

For 3^(40), 40 is also divisible by 5. Why don't we say 3^40 ends in 3?

Cydney Seigerman, Magoosh Tutor

Happy to clarify, Michael :)

When dealing with unit digit questions, we focus on the pattern of unit digits. In the case of 3, the unit digit follows a pattern that includes 4 terms: 3, 9, 7, 1. So, to calculate how many times we've gone through the pattern, we can divide the exponent by the number of terms. That's why we hone in on that 40 is divisible by 4 rather than 5. This shows us that to reach 3^43, we've gone through the pattern of unit digits 10 times. And we start the pattern over with 3 for the next exponent, 3^41. From there, as I explain above, we find that 33^43 falls in the third position and will therefore end in 1. :)

I hope this clears up your doubts!