number properties, 3 edition ,chapter 10 ,#11, page 179

[sowjanyaangara]

11) m is a positive integer less than 300 and has exactly two even and two odd prime factors in its prime factorization. What is the maximum possible value of m?

even prime number is 2 and how can we get another even prime factor here. I dont have any clues how to solve the problem, please explain

Thanks

[tommywallach]

Hey Sowjan,

This question is pretty tough, but your problem is more general. I think you don't understand the definition of the word factorization. To factorize is to break a number down into all of its prime factors (usually by using the factor tree), not all of its distinct prime factors.

For example, the prime factorization of 12 is: 2 * 2 * 3 = 12

Notice that there are two 2s in the factorization of 12.

So we need a number with only two even prime factors. As you noticed, the only option for that is 2.

2 * 2 = 4

This tells us the number in question is a multiple of 4. So the other two factors have to multiply to something less than 75 (a quarter of 300). We want the biggest possible product. 73 and 71 are prime, so they won't have two factors. 69 does have two prime factors (3 and 23), so there you go:

2 * 2 * 3 * 23 = 276

Hope that helps!

-t

[manigandaraja100]

Hi tommy,
your explanation helped.Thanks.btw you have told that "73 and 71 are prime, so they won't have two factors", but its actually they won't have two prime factors. isn't it ?

[tommywallach]

Yes. You can see from the end of that sentence that I was discussing prime factors.

Glad to help!

-t