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:
Q. What is the units digit of 57^{45}?
A) 1
B) 3
C) 5
D) 7
E) 9
Ans: 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: 7
Q. What is the units digit of 39^{61}?
Ans: 9
93 videos77 docs104 tests

Cyclicity & Factorial: Number System Doc  9 pages 
Cyclicity: Number Theory Video  03:24 min 
Difficult Units Digits Questions Doc  4 pages 
1. How do you find the unit digit of a large power? 
2. What is the significance of the unit digit in mathematics? 
3. Can the unit digit of a large power be zero? 
4. Are there any exceptions to finding the unit digit of large powers? 
5. What strategies can be used to find the unit digit of large powers? 
Cyclicity & Factorial: Number System Doc  9 pages 
Cyclicity: Number Theory Video  03:24 min 
Difficult Units Digits Questions Doc  4 pages 

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