Is the HCF of a and b an odd number? 1) HCF of (a+103) and (b+2) is an

Question

Is the HCF of a and b an odd number?
1) HCF of  (a+103)  and  (b+2)  is an even number
2)  (a+b)  and  (a-b)  are co-prime

LeoN88 · Answer

1. For HCF of (a+103) and (b+2) to be even we must have a power of two in either terms.

(a+103) is even iff a is odd
(b+2) is even iff b is even.

For two numbers- even and odd, their HCF can never be even. Sufficient.

2. (a+b) and (a-b) are co primes iff
one is even and other one is odd (a,b)
Check: 7,4-> 11 and 3. Their (7,3) HCF has to be odd. Sufficient. IMO  D  Either independently sufficient.

Further check: 4,6-> 10, 2 Not co prime
3,9 two odds: 12, 6-> Not co primes
3,7-> 10, 4, not co primes.

IanStewart · Answer

If the GCD of a+103 and b+2 is even, then a+103 and b+2 are each divisible by 2. But that means a is odd, so a is not divisible by 2, and the GCD of a and anything else must be odd. So Statement 1 is sufficient.

No one needs to know the meaning of the word "co-prime" for the GMAT, but two numbers are called "co-prime" or "relatively prime" in math if their GCD is 1. So Statement 2 tells us a+b and a-b have a GCD of 1. Now, if a and b shared some divisor, say d, then a+b would be the sum of two multiples of d, so would automatically be divisible by d. The same is true of a-b. So if a and b share a divisor d, then a+b and a-b would both share that divisor, and their GCD would be at least d. Since their GCD is 1, then a and b cannot share any divisor besides 1, and their GCD is thus 1, so Statement 2 is sufficient and the answer is D.

firas92 · Answer

1. (a+103) and (b+2) have an even number as their HCF

Since an even number cannot be a factor of an odd number, both (a+103) and (b+2) must be even

For (a+103) to be even, a must be odd and for (b+2) to be even, b must be even

Therefore a is odd and b is even and therefore the HCF of a and b must indeed be odd

1 is sufficient

2. (a+b) and (a-b) are co-prime which means their only common factor is 1. Notice that (a+b) and (a-b) must be either both even or both odd. If they were both even, they would have at least two common factors (1 and 2) and so (a+b) and (a-b) must be both odd.

For a+b to be odd, one of the two must be odd and the other, even

And again, an even and an odd number can only have an odd HCF

2 is sufficient

Answer is  (D)

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ZZhang · Answer

Can someone explain how statement 2 is sufficient?

If a = 10.5 and b = 0.5, then a + b = 11 and a - b = 10

11 and 10 are co-prime but the GCF of a and b is 0.5, which is not conceptually even or odd.

Question does not specify whether a and b are integers.

IanStewart · Answer

It's not an official question, and the solutions in the thread were assuming the question would have been written like an official problem. Any similar real GMAT question would declare that a and b are positive integers. There are, technically speaking, two different ways to define GCDs for fractions, so if we were allowing a and b to be non-integers, it wouldn't be clear how to answer the question anyway, but the GMAT would absolutely never test you on GCDs except when you're dealing with positive integers.

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Is the HCF of a and b an odd number? 1) HCF of (a+103) and (b+2) is an

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