If x < 20, How many distinct factors does odd number x ha

[GMATinsight]

If x < 20, How many distinct factors does odd number x have?

1) 16x is divisible by 24

2) 14x is not divisible by 15


Source: https://www.GMATinsight.com

Answer: Option C

[DrMaths]

Possible values: x = 1,3,5,7,9,11,13,15,17,19

Write values as products of prime factors:
x = 1,3,5,7,3^2,11,13,(3*5),17,19

1) (2^4)x is divisible by (2^3)*3 simplified becomes 2x is divisible by 3, so x is divisible by 3, so x =3,9,15
2) (2*7)x is not divisible by 3*5 simplified becomes x is not divisible by 3*5, so x is not 15

So (1) and (2) combined results in x = 3, 3^2, i.,e. there is only 1 distinct factor: 3
Therefore (1) and (2) are both required to be SUFFICIENT

[DrMaths]

If x < 20, How many distinct factors does odd number x have?

FROM STATEMENT 1:
16x = 2*2*2*2*x
24 = 2*2*2*3
So 16x/24 = 2x/3, means that x is divisible by 3
So x = 3, 9, 15, which gives 2 distinct factors: 3 and 5

FROM STATEMENT 2:
14x = 2*7*x
15 = 3*5*
So 14x/15 = 2*7*x/(3*5), means that for non divisibility, then x is not divisible by 15
So x = 1, 3, 5, 7, 9, 11, 13, 17, 19, which gives 8 distinct factors: 1, 3, 5, 7, 11, 13, 17, 19

Hence, using both statements, x = 3 or 9, meaning only 1 distinct factor: 3

Therefore either Statement A alone, or Statement B alone, or both statements combined will all generate an answer.
However, only both statements together reduce it to a single unquestionable factor.




1) 16x is divisible by 24

2) 14x is not divisible by 15


Source: https://www.GMATinsight.com

Answer: Option C