Factors of x^2

Question

x is divisible by 42. Which of the following numbers is definitely a factor of x2? (Choose
all that apply.)
a) 63 b) 33 c) 36 d) 8

Ans: a & c --> Why though?

Let me tell you my thinking:

Prime Factors of 42 are 2,3 & 7

That means that x definitely has 2,3,and 7 as factors.

Now x^2 will have = 2*2*3*3*7*7

Now for any answer choice to be a factor of x^2 there should be atleast one 2, one 3 and one 7 or max of two of each prime factor.

So considering choice a) 63 has 3*3*7 but no 2. So how come this can be a factor? Can someone please help?

katealpha · Answer

Factors of a number do not have to contain all the primes that make a number. 
3 is a factor of 12 and there is no 2 in 3. Just remember that factors cannot contain MORE of each prime factors than factored number (that's why, if value of x is 42, 8 = 2^3 is not a certain solution) also factor cannot contain any ADDITIONAL primes (again, if value of x=42, 33 is not a certain solution as it has an additional prime: 11)

pike · Answer

Prime factors of 42 = 2, 3, 7
x / 42 implies 
x = 2 * 3 * 7 * ...
x^2 = 2*2 * 3*3 * 7*7 * ...

Is 63 a factor?
63 = 3 * 3 * 7

Clearly, these are all factors of x^2. That is

x^2 = 63 * 7 * 2*2 * ...

This is only true for a and c.

Spidy001 · Answer

x = (2^1)(3^1)(7^1)(p)

x^2 = (2^2)(3^2)(7^2)(p^2)

from the above equation we can easily say that 2^2 *3^2 =36 
 also 3^2 *7 =63 are definitely factors of x^2

Hence answer is a & c.

vinayrsm · Answer

enigma - I think you got confused between a "factor" and a "multiple". Factors need not have all the prime numbers (with their exponent power) but Multiples need to have that. i.e for a number 21 (prime factors 3, 7): 1,3,7 and 21 are all factors of 21. and 21, 42, 63 .. are all multiples of 21

nikhilsamuel89 · Answer

x is divisible by 42.
This means that x contains atleast 2,3,and 7...may or maynot contain other factors. But we're sure that it contains atleast one 2, one 3, and one 7.
Hence, x^2 MUST contain atleast 
 two 2s, two 3s, and two 7s.
options:
 a) 63 = 7*3*3. Hence x^2 div. by 63 
 b) 33 = 11*3. 
We're sure that x^2 is div. by 3, but not sure whether div. by 11. May or may not.
 c) 36 = 2*3*2*3. x^2 is div. by 36 
d) 8 = 2*2*2. Here, there is an extra 2, which we're not sure whether it is present in x^2 or not.

Shoumik · Answer

your answer choice doesnt need to HAVE ALL THREE prime factors. At the same time, your answer choice cannot have a prime factor that is NOT 2, 3, and 7.

Factors of x^2

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