What is the units digit of 7^42 .

Question

I know I can just plug it in a calculator, but what is a faster way

Answer 1

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.

Answer 2

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.

Answer 3

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

Answer 4

Yes, that's correct!