Question
If one thousand 7s are multiplied together (e.g., 7 x 7 x 7 x 7 x ... x 7), what digit is in the ones place of the product
Answers
GPT-4o mini
To determine the digit in the ones place of the product of one thousand \( 7 \)s multiplied together (i.e., \( 7^{1000} \)), we need to analyze the pattern of the ones digits of the powers of \( 7 \).
Calculating the first few powers of \( 7 \):
- \( 7^1 = 7 \) (ones digit is \( 7 \))
- \( 7^2 = 49 \) (ones digit is \( 9 \))
- \( 7^3 = 343 \) (ones digit is \( 3 \))
- \( 7^4 = 2401 \) (ones digit is \( 1 \))
- \( 7^5 = 16807 \) (ones digit is \( 7 \))
We can see that the ones digits repeat in a cycle:
- \( 7^1 \) has a ones digit of \( 7 \)
- \( 7^2 \) has a ones digit of \( 9 \)
- \( 7^3 \) has a ones digit of \( 3 \)
- \( 7^4 \) has a ones digit of \( 1 \)
- \( 7^5 \) has a ones digit of \( 7 \) (and the cycle continues)
The pattern of ones digits is \( 7, 9, 3, 1 \) and it repeats every \( 4 \) numbers.
To find the ones digit of \( 7^{1000} \), we need to determine the position of \( 1000 \) in relation to the cycle by calculating \( 1000 \mod 4 \):
\[
1000 \div 4 = 250 \quad \text{with a remainder of } 0,
\]
Thus, \( 1000 \mod 4 = 0 \).
According to our cycle:
- \( 1 \) corresponds to \( 7 \)
- \( 2 \) corresponds to \( 9 \)
- \( 3 \) corresponds to \( 3 \)
- \( 0 \) corresponds to \( 1 \)
Since \( 1000 \mod 4 = 0 \), the ones digit of \( 7^{1000} \) corresponds to \( 1 \).
Therefore, the digit in the ones place of the product of one thousand \( 7 \)s is
\[
\boxed{1}.
\]
Calculating the first few powers of \( 7 \):
- \( 7^1 = 7 \) (ones digit is \( 7 \))
- \( 7^2 = 49 \) (ones digit is \( 9 \))
- \( 7^3 = 343 \) (ones digit is \( 3 \))
- \( 7^4 = 2401 \) (ones digit is \( 1 \))
- \( 7^5 = 16807 \) (ones digit is \( 7 \))
We can see that the ones digits repeat in a cycle:
- \( 7^1 \) has a ones digit of \( 7 \)
- \( 7^2 \) has a ones digit of \( 9 \)
- \( 7^3 \) has a ones digit of \( 3 \)
- \( 7^4 \) has a ones digit of \( 1 \)
- \( 7^5 \) has a ones digit of \( 7 \) (and the cycle continues)
The pattern of ones digits is \( 7, 9, 3, 1 \) and it repeats every \( 4 \) numbers.
To find the ones digit of \( 7^{1000} \), we need to determine the position of \( 1000 \) in relation to the cycle by calculating \( 1000 \mod 4 \):
\[
1000 \div 4 = 250 \quad \text{with a remainder of } 0,
\]
Thus, \( 1000 \mod 4 = 0 \).
According to our cycle:
- \( 1 \) corresponds to \( 7 \)
- \( 2 \) corresponds to \( 9 \)
- \( 3 \) corresponds to \( 3 \)
- \( 0 \) corresponds to \( 1 \)
Since \( 1000 \mod 4 = 0 \), the ones digit of \( 7^{1000} \) corresponds to \( 1 \).
Therefore, the digit in the ones place of the product of one thousand \( 7 \)s is
\[
\boxed{1}.
\]