guide 4 number properties question 18 pg 171

[nruthya.rajagopal]

hi ,
this is the question.....x2 is divisible by 40 and 75?if x has 3 distinctly prime factors,which of the following could be the value of x?


I was not able to understand the answer given.
why x must have atleast two 2s and one 5 in its prime factorization?
How is 5 redundant ?

please reply soon.

Thanks
Nruthya

[tommywallach]

Hey Nruthya,

The factor foundation rule states that any given number is divisible by any combination of the prime factors it is made up of. Similarly, whenever you multiply two numbers together, the resultant number has ALL the prime factors that the individual numbers had (there is no overlap, in that case).

x^2 = x * x --> so we need to get the factors of x^2 out of those x's.

If x ^ 2 is divisible by 40, that means it has prime factors: 2, 2, 2 and 5.

Those had to come out of the individual x's. Now, if each x had only a single 2 in its prime factors, x^2 would only have two 2s in its prime factors. But we need three 2s. To get that, each x must have two 2s. Yes, this does mean you'd end up with four, but that's okay.

Think of it this way, if you have $50, you don't have $60, but you DO have $40. Similarly, if you have four 2s, you don't have five 2s, but you DO have three 2s.

Now, the 75 tells us that x^2 has one 3 and two 5s. That means each x must have one three in it (because you couldn't have HALF a 3), and each x must have one five in it.

You might be thinking that we have three 5s altogether (one from the 40, and two from the 75), but there is overlap.

Here's a simple example. Imagine I told you that x is divisible by both 4 and 8. Now, at first glance, you might think that the 4 gives us two 2s, and the eight gives us three 2s, for a total of five 2s. But logically, we know that the smallest number divisible by both 4 and 8 is 8, which only has three 2s.

This is because there is overlap. You only take the GREATEST number of each individual prime factor that you find, and ignore any smaller numbers. In the case of 4 and 8, we get two 2s and three 2s, so we ignore the two 2s. In the case of 40 and 75, we get one 5 and two 5s, so we ignore the one 5.

Let me know if that all makes sense!

-t

[nruthya.rajagopal]

thanks.
i understood it clearly.

[tommywallach]

Glad to help!

-t