What is the units digit of 7^42 .
I know I can just plug it in a calculator, but what is a faster way
The cycle: 7^1
7^2=..9
7^3=..3
7^4=..1
7^5=..7
=> Every 4th power starts a new cycle of digits
Divided the index number of 7^42 by 4 you have the remainder 2
=> The last digit is 9
Yes, that's correct!
Oh, calculating the units digit of 7^42? Let me put on my mathematics clown nose for this one!
Now, let's see. We know that the units digit of 7^1 is 7, and the units digit of 7^2 is 9. So, it seems like the units digit of the powers of 7 alternate between 7 and 9.
Since 42 is an even number, the units digit of 7^42 must be the same as the units digit of 7^2, which is... *drumroll*... 9! Hope that brings a smile to your face!
To find the units digit of a number raised to a power, you can use patterns and properties of numbers. In this case, let's observe the pattern of the units digits when 7 is raised to different powers:
7^1 = 7
7^2 = 49 (the units digit is 9)
7^3 = 343 (the units digit is 3)
7^4 = 2401 (the units digit is 1)
7^5 = 16807 (the units digit is 7)
...
From this pattern, we can see that the units digit of 7 repeats in a cycle of 4: 7, 9, 3, 1. This means that any power of 7 that is a multiple of 4 will have a units digit of 1.
Now, let's find the units digit of 7^42. Since 42 is divisible by 4, the units digit of 7^42 will be the same as the units digit of 7^4, which is 1.
Therefore, the units digit of 7^42 is 1.
You only need to calculate the first digits of every number because when you multiply a number again and again the latter digits don't affect the first one.
To start 7*7 = 49 but we only keep the 9
9*7 = 63 keep the 3
3*7 = 21 keep the 1
1*7 = 7
Notice that we end up with 7 again.
If we keep going we will get 7,9,3,1 over and over again.
7 to power of any multiple of four (7^4n start with 7^0) will get you a 1.
If you multiply once more you will get a 7 then a 9 then 3 then back to a 1.
42 is 2 away from a power of 4
40 will be 1, 41 will be 7, and 42 will be 9
9 is the answer.
start listing the powers of 7. Just the last digits are important:
7
49
343
...1
...7
Every 4th power starts a new cycle of digits.
40 = 4*10, so the final digit of 7^40 is 1.
Now you know how to get the final digit of 7^42.