Is the product of two integers A and B odd?

Hi ggurface ,

Is it necessary to know that # of factors in perfect square is always odd?

let's say we have a number 84700, and we need to know the number of factors it has,
Then - 84700 = 2^2 * 5^2* 7^1* 11^2

so in order for us to determine number of factors 84700 has we would add 1 to each power and then would add them all together.

(2+1)+(2+1)+(1+1)+(2+1)= 3+3+2+3=11 factors, now we know 84700 has 11 factors in it.

Coming to your question- statement 1 says " A is the number of factors of N", where N is a perfect square"
It clearly means we need to know whether A is even/odd.

N^even +1 = odd=A
I figured out in above mentioned method.

I don't know if there's any other method exists by which the problem could be solved. There might be.Please comment GMAT Experts if i'm wrong in my explanation.

but the only way for AB to be odd is if A & B are BOTH ODD, and if B = A^3 - 1 then doesnt that mean they always differ in even/odd?

Part 2 of this question says " A is a product of two consecutive prime numbers...."

the only two consecutive prime numbers are 2 and 3, so why even consider the possibility of 3*5 or 5*7??

Please tell if you agree.

Damm, it really is a knockout question

No of factors in a perfect square is odd, therefore B would be even
Test it
Say 36....no of factors is A = 9, then B = 9^3 - = even integer
Therefore odd x even must be even, so the answer is sufficiently no .........Sufficient

Stmt 2. A could be 2*3 = even or, 3*5 = odd
If A is even, then B must be even for (B+ 3^11) to be odd, and for the total sum of A+ (B+ 3^11) to be odd, therefore A*B is even

If A is odd, then B must be odd for (B + 3 ^11) to be even, and for the total sum of (B+ 3^11), to be odd, therefore A*B is odd
Stmt 2 is insufficient

Gosh, I need to get neater in my paperwork

Detailed Solution

Step-I: Given Info

The question tells us about two integers \(A\) & \(B\) and asks us if their product is odd.

Step-II: Interpreting the Question Statement

To find if the product of two numbers is odd/even, we need to establish if either of the number is even or not. In case either of the number is even, the product would be even else the product would be odd.

Step-III: Statement-I

Statement- I tells us that \(A\) is the number of factors of a perfect square, since the no. of factors of a perfect square is odd, we can deduce that \(A\) is odd. Since \(A\) is not even, to find the nature of product of \(A\) & \(B\), we need to find if \(B\) is odd/even.

It’s given that \(B= A^3 -1\), since we have established that \(A\) is odd, \(A^ 3\) will also be odd. Subtracting 1 from an odd number will give us an even number, hence we can deduce that \(B\) is even.
Since, we know now that \(B\) is even it is sufficient for us to deduce that the product of \(A\) & \(B\) would be even.

Thus Statement-I is sufficient to get the answer.

Step-IV: Statement-II

Statement- II tells us that \(A\) is a product of two consecutive prime numbers, so \(A\) may be even if one of the prime number is 2 and may be odd if none of the prime number is 2. So, we can’t establish with certainty the even/odd nature of \(A\).
The statement also tells us that \(A+ B + 3^{11} =\) odd, since 3 is an odd number, \(3^ {11}\) would also be odd and subtracting an odd from an odd number would give us an even number. So, we can rewrite
\(A+ B =\) even which would imply that either both \(A\), \(B\) are even or both are odd. In both the cases the nature of product of \(A\) & \(B\) can’t be established with certainty, it will be even if both \(A\) & \(B\) are even and will be odd if both \(A\) & \(B\) are odd.

So, Statement- II is not sufficient to answer the question.

Step-V: Combining Statements I & II

Since, we have a unique answer from Statement- I we don’t need to be combine Statements- I & II.
Hence, the correct answer is Option A

Key Takeaways

1. In even-odd questions, simplify complex expressions into simpler expressions using the properties of even-odd combinations.
2.The number of factors of a perfect square would always be odd.
3.Know the properties of Even-Odd combinations to save the time spent deriving them in the test

Regards
Harsh

Dear riyazgilani

Please note that there is a difference between 'consecutive numbers that are prime' and 'consecutive prime numbers.'

Our concept file mentions that 2 and 3 are the only 'consecutive numbers that are prime'

On the other hand, examples of 'consecutive prime numbers' are many: 2, 3, 5, 7, 11, 13, 17, 19 . . . all these prime numbers are consecutive prime numbers, because they come one after the other in the list of prime numbers.

So, when this question says that two numbers are consecutive prime numbers, they can be (2,3) or (5, 7) or (7,11) etc.

I hope this clarification helped.

Best Regards

Japinder

Dear riyazgilani

Please note that there is a difference between 'consecutive numbers that are prime' and 'consecutive prime numbers.'

Our concept file mentions that 2 and 3 are the only 'consecutive numbers that are prime'

On the other hand, examples of 'consecutive prime numbers' are many: 2, 3, 5, 7, 11, 13, 17, 19 . . . all these prime numbers are consecutive prime numbers, because they come one after the other in the list of prime numbers.

So, when this question says that two numbers are consecutive prime numbers, they can be (2,3) or (5, 7) or (7,11) etc.

I hope this clarification helped.

Best Regards

Japinder

Is the product of two integers \(A\) and \(B\) odd?

(1) \(A\) is the number of factors of \(N\), where \(N\) is a perfect square and \(B = A^3 - 1\)
(2) \(A\) is a product of two consecutive prime numbers and when \(B + 3^{11}\) is added to \(A\), the sum is an odd number.

Source: e-GMAT

You should read more carefully: A is a product of two consecutive prime numbers. 3 and 5 ARE consecutive primes.

Why are we not considering 1² = 1, 1 as a perfect square? Is 1 not generally considered as a perfect square

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