Raising to a power is iterated multiplication. Luckily, you can find your units digit with a simple multiplication pattern, even when you’re working with large powers. (For a refresh of the multiplication rules for unit digits, see our post on difficult units digits.)
See how you do with this question:
What is the units digit of 57^{45}?
A) 1
B) 3
C) 5
D) 7
E) 9
To solve this, we’ll begin examining smaller powers and look for a pattern.
57^{1} = 57 (the units digit is 7)
57^{2} = 3,249 (the units digit is 9)
57^{3} = 185,193 (the units digit is 3)
Aside: Since these powers increase quickly, it’s useful to notice that we need only multiply the units digit each time. For example, the units digit of 57^{2} is the same as the units digit of 7^{2}. Similarly, the units digit of 57^{3} is the same as the units digit of 7^{3}.
So, once we know that the units of 57^{2} is 9, we can find the units digit of 57^{3} by multiplying 9 by 7 to get 63. So the units digit of 57^{3} is 3.
To find the units digit of 57^{4}, we’ll multiply 3 by 7 to get 21. So the units digit of 57^{4} is 1.
When we start listing the various powers, we can see a pattern emerge:
The units digit of 57^{1} is 7
The units digit of 57^{2} is 9
The units digit of 57^{3} is 3
The units digit of 57^{4} is 1
The units digit of 57^{5} is 7
At this point, we should recognize that the pattern begins to repeat. The pattern goes: 793179317931…
Since the pattern repeats itself every 4 powers, we say that the “cycle” equals 4
Now comes an important observation:
As you can see, since the cycle = 4, the units digit of 57k will be 1 whenever k is a multiple of 4.
Now to find the units digit of 5745, all we need to do is recognize that the units digit of 5744 is 1 (since 44 is a multiple of 4).
From here, we’ll just continue with our pattern:
The units digit of 57^{44} is 1
The units digit of 57^{45} is 7
The units digit of 57^{46} is 9
The units digit of 57^{47} is 3 . . . etc.
So, the units digit of 57^{45} is 7, which means the answer is D.
Practice Question:
Q. What is the units digit of 83^{75}?
Ans: 6
Q. What is the units digit of 39^{61}?
Ans: 9
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