Question

Without doing the actual long division process, find if the expansion of the rational number \( \frac{637}{7280} \) is a terminating or non-terminating repeating decimal. Give reasons for your answer.

Hint:

A rational number's decimal expansion terminates if and only if the prime factorization of its denominator (after simplifying) contains only 2 and 5.

Answer 1

To determine whether the decimal expansion of \( \frac{637}{7280} \) is terminating or non-terminating repeating, we must first find the prime factorization of the denominator. Step 1: Prime factorization of 7280 Divide 7280 by the smallest prime numbers: \[ 7280 \div 2 = 3640 \quad \text{(divide by 2)} \] \[ 3640 \div 2 = 1820 \quad \text{(divide by 2 again)} \] \[ 1820 \div 2 = 910 \quad \text{(divide by 2 again)} \] \[ 910 \div 2 = 455 \quad \text{(divide by 2 again)} \] Now, divide by 5 (since 455 is divisible by 5): \[ 455 \div 5 = 91 \quad \text{(divide by 5)} \] Next, divide by 7 (since 91 is divisible by 7): \[ 91 \div 7 = 13 \quad \text{(divide by 7)} \] 13 is a prime number, so we stop here. Therefore, the prime factorization of 7280 is: \[ 7280 = 2^4 \times 5 \times 7 \times 13. \] Step 2: Determine if the decimal expansion is terminating or repeating For the decimal expansion of a rational number to terminate, the prime factorization of the denominator (after simplifying the fraction) must contain only the primes 2 and 5. Since 7280 has the prime factors 2, 5, 7, and 13, it contains primes other than 2 and 5. Therefore, the decimal expansion of \( \frac{637}{7280} \) is non-terminating repeating.
Conclusion:
The decimal expansion of \( \frac{637}{7280} \) is non-terminating repeating.