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Excellent questions, Priti (not only those two - in general)
1. I agree with ranga41 on a tricky B
Statement (1) tells us that n is a prime number, so could be 2,3,5, so on. Not sufficient.
Statement (2)... I am evaluating two sets: all the prime numbers and all the non-prime integers >2
Among the prime numbers only 2 fulfills the stated condidtion, because
3-1=2 5-1=4 and so on...
All the non-prime integers>2 can be broken down to primes, so the simplest of them will have at least 4 factors: 1, x, y and xy, where x and y are primes.
If x and y are even, than xy - 1=odd, but y-x is even => not ok
If x is even and y is odd, than xy-1=odd, but y-1 is even => not ok
If x is odd and y is even, same as above => not ok
If x and y are odd, than all the differences are even => not ok
Since we have run out of numbers to test, 2 seems to be the only number which satisfies statement (2), so the answer should be B.
I will go with a tentative B on the second question. Here is my reasoning:
Statement (1) gives us information about x, but nothing about y => not sufficient.
Statement (2) tells us that 12 is a factor of y. If we substitute into the stem, we get
So we need to find the greatest common divisor of x=12(8z+1) and y=12z.
12 is obviously a commont divisor, but what about (8z+1) and z? Тheir only common divisors are 1 and -1, because (8z+1)/z=8+1/z.
To sum it up, I think the second statement gives us enough information and I pick answer choice B
1. n has exactly 2 positive factors. 2. The difference of any 2 distinct positive factors of n is odd.
1. n has exactly 2 factors, that would be 1 and n itself. So I agree that any primary number would fit the bill. Insuficient.
2. Since 1 and n would be two of the factors. Therefore n-1 is odd. In other words, n must be even. But if n is an even number that is greater than 2, then 2 must be its factor too. And n-2 would NOT be odd. In other words, n has to be an even number that is not greater than 2. That is, n=2. Sufficient.
I'm choosing B.
If x and y are +ve integers such that x = 8y + 12, what is the greatest common divisor of x and y?
1. x = 12u, where u is an integer. 2. y = 12z, where z is an integer.
=> 2/3y is an integer.
In other words y is divisible by 3. Since x is also divisible by 3, 3 is one of the common divisors of x and y. However, is it the greatest? We can't tell. For example, if y is divisible by 2, then 6 could be one of the common divisors. And we still don't know if it is the greatest.
Since 1 would be the greatest common divisor of z and (8z+1), we know for sure that the greatest common divisor of x and y is 12.
Keep on asking, and it will be given you;
keep on seeking, and you will find;
keep on knocking, and it will be opened to you.