Question

This is from my ACT prep guide workbookI am a beginner at this kind of math If you can, please explain step-by-step thoroughly on how to answer this problem :) thank you in advance

Answer 1

ANSWER

J. 63

EXPLANATION

We know that x is a positive integer and the other expression is also a positive integer. First, simplify the expression by taking x as a common factor,

$\frac{x}{3}+\frac{x}{7}+\frac{x}{9}=x(\frac{1}{3}+\frac{1}{7}+\frac{1}{9})$

Then, add the coefficients. First, we have to find the least common denominator. 9 is a multiple of 3, and 7 is a primal number, thus the least common denominator is 9x7 = 63,

$x(\frac{1}{3}+\frac{1}{7}+\frac{1}{9})=x\frac{21+9+7}{63}=x\cdot\frac{37}{63}$

37 and 63 have no common factors, so that fraction cannot be simplified. As we can see there is a fraction multiplying x, but we were told that the expression was a positive integer, so we have to find x so that this expression results in an integer. Also, x must be an integer too. The least value of x is the one that cancels out the denominator of the fraction,

$x\cdot\frac{37}{63}=63\cdot\frac{37}{63}=37$

Hence, the least value of x is 63