[email protected] wrote:If r and s are positive integers, can the fraction r/s be
expressed as a decimal with only a finite number of nonzero digits?
(1) s is a factor of 100.
(2) r is a factor of 100.
A TERMINATING decimal has a FINITE NUMBER OF DIGITS:
.5
.123
.8730253.
A NON-TERMINATING decimal has an INFINITE NUMBER OF DIGITS:
.33333....
.121212....
.871871871...
To determine whether a fraction will yield a terminating decimal:
1. Put the fraction in its MOST REDUCED FORM.
2. PRIME-FACTORIZE the denominator.
If the prime-factorization of the denominator includes ONLY 2'S AND/OR 5'S, the fraction will yield a TERMINATING decimal.
If the prime-factorization of the denominator includes ANY OTHER PRIME NUMBER, the fraction will yield a NON-TERMINATING decimal.
Case 1: 3/120
In its most reduced form, 3/120 = 1/40.
40 = 2² * 5.
Since the the prime-factorization of the denominator includes only 2's and 5's, 3/120 will yield a TERMINATING DECIMAL:
3/120 = .025
Case 2: 15/110
In its most reduced form, 15/110 = 3/22.
22 = 2*11.
Since the prime-factorization of the denominator includes 11 -- a prime number OTHER THAN 2 OR 5 -- 15/110 will yield a NON-TERMINATING DECIMAL:
15/110 = .1363636...
Onto the problem at hand.
Question rephrased: When r/s is in its most reduced form, will the prime-factorization of the denominator include only 2's and/or 5's?
Statement 1: s is a factor of 100
Since 100 = 2²5², the prime-factorization of s cannot include any prime number other than 2 and/or 5.
Thus, the value of r is IRRELEVANT:
Since the prime-factorization of s cannot include any prime number other than 2 and/or 5, the denominator of r/s in its most reduced form cannot include any prime number other than 2 and/or 5.
SUFFICIENT.
Statement 2: r is a factor of 100
No information about s.
If r=1 and s=2, then r/s = 1/2 = .5.
In this case, r/s is terminating.
If r=1 and s=3, then r/s = 1/3 = .33333...
In this case, r/s is NOT terminating.
INSUFFICIENT.
The correct answer is
A.
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