Number Properties of Primes

[Cedagmat]

81. If n is an integer greater than 1, and n is not a prime number, then which of the following must be true?


(A) n is the sum of three prime numbers
(B) n is the difference between 2 even numbers
(C) n is the difference between one even number and one odd number
(D) n is the product of one even number and one odd number
(E) n is the product of prime numbers

OA[spoiler]: E[/spoiler]
I initially picked D as my answer since the first four numbers that meet the condition would be 4, 6, 8, 9. I don't think the question is specific enough--but perhaps someone can point out where I am wrong here.


86. If 5x^2 has two different prime factors, at most how many different prime factors does x have?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

[spoiler]OA: B[/spoiler]

I took this to mean, since 5x^2 has two prime factors, they must be 5 and x^2. Thus x^2 is prime. Thus x would only have one prime factor. What is wrong with my reasoning here?

87. How many positive integers, from 2 to 100, inclusive, are not divisible by odd integers greater than 1?
(A) 5
(B) 6
(C) 8
(D) 10
(E) 50

[spoiler]OA: B[/spoiler]
I attempted to first find all the different integers divisible by 3, 5, 7, 9, and 11 but this was pretty time consuming. Any other ideas?

[Brian@VeritasPrep]

Hey, Cedagmat - good questions! The number properties stuff is incredibly important. Let me tackle a few:


1) Here, E is really the definition of a non-prime positive integer. If it's not prime, you can express it as the product of primes, so you should get comfortable with that definition so that you don't have to plug in.

For D, keep in mind that 9 - the fourth number in your list - is NOT the product of even*odd. With integers, the only way you can have an odd product is if all factors are odd - the minute you add an even number to the mix then you get an even product.

I'd also advise this for the "must be true" problems - when picking numbers, your goal should be to prove that it "could be false". Instead of just looking at the first four numbers, try to create a number that is not a product of an even and an odd - try two odds or two evens to see what you get. Since 1 is an odd number, the two evens don't work, but if you try 9 the only way you can get it is 1*9 and 3*3.


2) Here's another strategic point along with the "must be true" tip above - if they ask for "at most", your goal is to maximize the number of prime factors. You need to try to find ways to add another prime factor if at all possible, and that 5 is a clue. If that entire term, 5x^2 has two prime factors, and we know that one of them is 5, then how can we get another prime factor for x other than the one prime factor that 5x^2 has left? We can also include a 5 in x (say, x = 10) so that 5x^2 has only two prime factors (5 and 2), but x gets its second prime factor out of a duplicate 5.


3) Here, think about what makes a number ONLY divisible by even factors. If no odd factors apply, then you'd have to eliminate 6, 10, 12, etc. because 6 is 3*2; 10 is 5*2; 12 is 3*2*2... What numbers remain? 2 and 4 in that 2-12 range, and that's because the only thing that divides into them are prime factors of 2. Any odd number is out and any even number divisible by 3, 5, 7, 11, etc. are out. So if we can only have "purely" 2-based numbers, that leaves:

2
2^2 = 4
2^3 = 8
2^4 = 16
2^ 5 = 32
2^6 = 64

and then anything else that ONLY has twos as a factor is too high, so there are 6 numbers that are solely even-based.

[goyalsau]

Real Good question's,
What is the source.


If n is an integer greater than 1, and n is not a prime number, then which of the following must be true?


n is the difference between one even number and one odd number or n is the difference between one odd number and one even number.

I just want to know for the sake of knowledge, Is this definition right for any integer that is greater than 1 and not prime....

Am i covering all the possible values of n

[goyalsau]

2) Here's another strategic point along with the "must be true" tip above - if they ask for "at most", your goal is to maximize the number of prime factors. You need to try to find ways to add another prime factor if at all possible, and that 5 is a clue. If that entire term, 5x^2 has two prime factors, and we know that one of them is 5, then how can we get another prime factor for x other than the one prime factor that 5x^2 has left? We can also include a 5 in x (say, x = 10) so that 5x^2 has only two prime factors (5 and 2), but x gets its second prime factor out of a duplicate 5.

Can you please explain this one,
what if we take the value of x as 3 , 5x^2 will still have only 2 prime factors.
I am confused with this one...