How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7? (1point).
The repeating cycle of a repeating decimal depends on the prime factors of the denominator. In this case, the denominator is 7, which is a prime number.
When a rational number has a prime number as the denominator, the repeating cycle of the decimal representation will be equal to one less than the denominator. Therefore, the repeating cycle of a decimal with a denominator of 7 will be 7 - 1 = 6 digits long.
To determine the length of the repeating cycle of a decimal representation, we need to find the smallest positive integer (let's call it "n") such that 10^n (10 raised to the power of n) is congruent to 1 modulo 7.
Let's calculate the remainders of 10^n for increasing values of n:
10^1 ≡ 3 (mod 7)
10^2 ≡ 2 (mod 7)
10^3 ≡ 6 (mod 7)
10^4 ≡ 4 (mod 7)
10^5 ≡ 5 (mod 7)
10^6 ≡ 1 (mod 7)
We see that for n = 6, 10^n is congruent to 1 modulo 7. This means that the decimal representation of any rational number with a denominator of 7 will have a repeating cycle of length 6.
So, the repeating cycle of a repeating decimal with a denominator of 7 will be 6 digits long.