Rational Numbers
Given A Rational Number , Write It As A Fraction In Lowest Terms and Calculate the Product of the Resulting Numerator and Denominator . For How Many Rational Numbers Between and Will Be the Resulting Product ?
Question from AIME.
The answer is 128.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
if the fraction is in the form p/q , then p<q and gcd(p,q) = 1 (that means greatest common divisor of p and q is 1) . prime numbers less than 20 are 2,3,5,7,9,11,13, 17, 19 .you can easily see that , there are 8 prime numbers less than 20 and each of them can be a factor of one of p or q . combination of numbers for p can be selected in 2^8 ways . but since p<q , only half of them will be between 0<(p/q)<1 . so , desired solution is , 2^8 /2 = 128