7^0 = 1
7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
Pattern of ones digit is 1,7,9,3,1,7,9,3...
However, I don't understand how you can find this pattern in the values given above?
What the patters is supposed to show is..
in 537^102 you only have to look at what happens to the ones digit
e.g.
537^0 = 1 vs 7^0 = 1
537^1 = 537 vs 7^1 = 77
537^2 = 288369 vs 7^2 = 49
537^3 = 157854153 vs 7^3 = 343
537^4 = 8315....1 vs 7^4 = 2401
notice the last digit is either
1 , 7, 9, or 3 , and cycles in that pattern
If the exponent is odd ---> not our problem
if the exponent is even and divides by 4, then it end in a 1
if the exponent is even but does not divide by 4, it ends in a 9
our exponent of 102 is even but not divisible by 4, so
537^102 will end in a 9
Thank you so much! Your explanation was very helpful!
the 2nd line of the e.g.'s should say
537^1 = 537 vs 7^1 = 7
7 xxxxx
Also, the originally problem is finding the ones digit 537^102
and I found out that 7 was a one digit factor
To understand the pattern of the ones digit in the values given above, it is important to observe the cyclic behavior that occurs.
When calculating the powers of 7, the ones digit of each result depends only on the ones digit of the base (7) and its position in the sequence.
In this case, we can notice that the ones digit goes through a cycle of 4 digits: 1, 7, 9, and 3. The cycle is repeated for each subsequent power of 7.
To find the pattern, calculate the ones digit of consecutive powers of 7:
- 7^0 = 1
- 7^1 = 7
- 7^2 = 49 (The ones digit is 9)
- 7^3 = 343 (The ones digit is 3)
- 7^4 = 2401 (The ones digit is 1)
From this point onward, the sequence repeats:
- 7^5 = 16807 (The ones digit is 7)
- 7^6 = 117649 (The ones digit is 9)
- 7^7 = 823543 (The ones digit is 3)
- 7^8 = 5764801 (The ones digit is 1)
So, the pattern is 1, 7, 9, 3, and it continues in the same order as you keep increasing the exponent.
Thus, if you need to find the ones digit of any power of 7, you can divide the exponent by 4 and look for the remainder. The remainder will correspond to the position in the cycle, which will give you the pattern.