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Re: m is a positive integer less than 300 and has exactly two e [#permalink]
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How does something have two even prime factors? Is 2 not the only even prime factor? Every other even number is divisible by 2.
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Re: m is a positive integer less than 300 and has exactly two e [#permalink]
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ajudd22 wrote:
How does something have two even prime factors? Is 2 not the only even prime factor? Every other even number is divisible by 2.


Hi, that's how GRE put a trap ques.

Plz see the ques "m is a positive integer less than 300 and has exactly two even and two odd prime factors in its prime factorization."

The sentence marked as bold, provides you a criteria. Your approach should satisfy the criteria. i.e the number must have 2 even prime factors and 2 odd prime factors
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Re: m is a positive integer less than 300 and has exactly two e [#permalink]
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ajudd22 wrote:
How does something have two even prime factors? Is 2 not the only even prime factor? Every other even number is divisible by 2.


I think you are having a confusion between prime numbers and prime factorization

Here more on number properties of the numbers on our math book

https://gre.myprepclub.com/forum/gre-quant ... tml#p51913

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Re: m is a positive integer less than 300 and has exactly two e [#permalink]
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why it can't be 288?
it has two even and two odd prime factor...
288=2*2*3*3*8
and for the record, question don't mention prime factor has to be distinct....
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Re: m is a positive integer less than 300 and has exactly two e [#permalink]
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void wrote:
why it can't be 288?
it has two even and two odd prime factor...
288=2*2*3*3*8
and for the record, question don't mention prime factor has to be distinct....

288 = (2)(2)(2)(2)(2)(3)(3)
So, 288 has five (not two) even primes (2's) in its prime factorization.
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Re: m is a positive integer less than 300 and has exactly two e [#permalink]
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So two even primes means that the number must be a multiple of 4 and must be less than 300. This means the number must be less than 300/4 = 75.
Now we need to find the product of two primes closest to 75. It comes out to be 3*23=69.
Hence the number is 4*69 = 276
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Re: m is a positive integer less than 300 and has exactly two e [#permalink]
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Carcass wrote:
OE

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Because the only even prime number is 2, m must be the product of 2 × 2 = 4 and two odd primes. Further, due to the restriction that m be less than 300, the product of those two odd primes must be less than \(\frac{300}{4} = 75\). Thus, you need to find the largest number smaller than 75 that is the product of exactly two odd prime numbers. You can rule out even numbers, and you have 73, which is prime, as is 71; 69 is the product of two prime numbers (3 and 23). Thus, the maximum possible value for m is 2 × 2 × 3 × 23 = 276.



A slightly faster was to get to 69. Once you know that you need to find the largest number smaller than 75 that is the product of exactly two odd prime numbers, it's fair to guess one of the prime factors is 3.* What is the other prime factor? The largest prime number smaller than 75/3, aka largest prime < 25. That's 23. 23*3 is 69.

Multiply 69 by 4, and you get the answer.

*This is just an intuition I have, but I checked it against other possibilities and it seemed to be true. Like, if I guessed the prime factor was 5 or 13, that wouldn't work. Could someone explain why 3 is a reasonable guess here? My guess is something to do with, you want to find a number whose product is close to 75, and generally the product of two numbers far apart on the number line is going to be more than the product of numbers closer? But that seems wrong? It seems like there's something generalizable here.
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