Archit143 wrote:
If n = (33)^43 + (43)^33 what is the units digit of n?
A. 0
B. 2
C. 4
D. 6
E. 8
This is a units digit pattern question. The first thing to recognize is that in units digit pattern questions we only care about the units digit place value. Thus, we can rewrite the problem as:
(3)^43 + (3)^33
We now need to determine the units digit of (3)^43 + (3)^33. Let's determine the pattern of units digits that we get when a base of 3 is raised to consecutive exponents.
3^1 = 3
3^2 = 9
3^3 = 27 (units digit of 7)
3^4 = 81 (units digit of 1)
3^5 = 243 (units digit of 3)
Notice at 3^5, the pattern has started over:
3^6 = units digit of 9
3^7 = units digit of 7
3^8 = units digit of 1
So we can safely say that the base of 3 gives us a units digit pattern of 3, 9, 7, 1, 3, 9, 7, 1, …) that repeats every four exponents. Also notice that every time 3 is raised to an exponent that is a multiple of 4, we are left with a units digit of 1. This is very powerful information, which we can use to solve the problem. Let’s start with the units digit of (3)^43.
An easy way to determine the units digit of (3)^43, is to find the closest multiple of 4 to 43, and that is 44. Thus we know:
3^44 = units digit of 1
So we can move back one exponent in our pattern and we get:
3^43 = units digit of 7
Let’s now determine the units digit of (3)^33.
We already know that the pattern of units digits for powers of 3 will be 3, 9, 7, 1, 3, 9, 7, 1, … An easy way to determine the units digit of (3)^33 is to find the closest multiple of 4 to 33, and that is 32. Thus we know:
3^32 = units digit of 1
So we can move up one exponent in our pattern and we get:
3^33 = units digit of 3
The last step is to add the two units digits together so we have:
7 + 3 = 10, which has a units digit of zero)
Answer is A.
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