[email protected] wrote:Q: IS THE POSITIVE INTEGER X A PRIME NO.
A: 3X + 1 IS A PRIME NO
B: 5X + 1 IS A PERFECT SQUARE
Hi
[email protected]!
Statement (1):
There are 2 basic cases possible: either 3X+1=2, or 3X+1= any other (odd) prime.
If 3X+1=2
3X = 1
X = 1/3, BUT, X is an integer so this case is NOT possible, so it must be the case that 3X+1=odd prime. Let's see what values are possible when we solve that for X:
3X+1 = odd prime
3X = odd prime - 1 (so, because an odd-1 is an even)
3X = even
X = even.
So X will be an even number, because there is one prime even number we must test that:
Let X=2, then 3X+1 = 3(2)+1 = 6+1 = 7 (PRIME), so 3X+1 is prime when X=2 (prime) so the answer to the original question is YES, X is Prime!
Let X=4, then 3X+1 = 3(4)+1 = 12+1 = 13 (PRIME), so 3X+1 is prime when X=4 (non-prime) so the answer to the original question is NO, X is NOT Prime!
We have 2 different answers from statement (1), so NOT sufficient!
Statement (2):
Now for the tougher statement - if 5X+1=perfect square, let's take a look at what this means for X. Call the perfect square Y^2.
5X+1 = Y^2
5X = Y^2 - 1
X = (Y^2 - 1)/5
So, we can find values of INTEGER X by finding perfect squares that are 1 more than a multiple of 5 (so that they will divide evenly). Let's list the perfect squares up to 12 (since we should have those memorized):
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
From this list, 16, 36, 81 and 121 are the ones that are 1 more than a multiple of 5. So
X = (16-1)/5 = 15/5 = 3 Prime!
X = (36-1)/5 = 35/5 = 7 Prime!
X = (81-1)/5 = 80/5 = 16 NOT Prime!
We showed two different answers from statement (2) so NOT sufficient!
Statement (1+2):
From statement (1): Integer X is even
From statement (2): Integer X could be several numbers, some even and some odd, but the smallest values is 3.
Putting this together, X is an even number but cannot be 2 so we have our answer - Integer X is NOT a Prime number, therefore, the statements together are Sufficient!
The correct answer is
C.
I hope this helps!
Whit
