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If a and b are positive integers, what is the greatest common divisor of a, b, and 3a + 23b?
(1) a = 4
(2) The greatest common divisor of a, b, and 23a + 3b is 4.
(1) a = 4
(2) The greatest common divisor of a, b, and 23a + 3b is 4.
27 Apr 2022, 20:22
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If a and b are positive integers, what is the greatest common divisor of a, b, and 3a + 23b?
(1) a = 4
Nothing about other numbers.
But we can say that the greatest common divisor can be 4 or its factors, that is 1 , 2 or 4.
(2) The greatest common divisor of a, b, and 23a + 3b is 4.
we can have different answers.
But the GCD will surely be 4 or more as both a and b are multiples of 4.
Say a=4b, then GCD of 4b, b and 23*4b+3b will be b, and b could be anything.
Similarly if b=4a, then it would depend on a.
Combined
The common integer in two statements is 4.
Sufficient
(1) a = 4
Nothing about other numbers.
But we can say that the greatest common divisor can be 4 or its factors, that is 1 , 2 or 4.
(2) The greatest common divisor of a, b, and 23a + 3b is 4.
we can have different answers.
But the GCD will surely be 4 or more as both a and b are multiples of 4.
Say a=4b, then GCD of 4b, b and 23*4b+3b will be b, and b could be anything.
Similarly if b=4a, then it would depend on a.
Combined
The common integer in two statements is 4.
Sufficient
25 Aug 2022, 09:12
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(1) - Since we don't know anything about the value of b, statement 1 is not sufficient.
(2)- Since the GCD of a, b, 23a+3b is 4, we can conclude that either a or b is equal to 4 and both are divisible by 4. Now, since a and b are divisible by 4, 3a+23b will also be divisible by 4(the product of a multiple of 4 and any other multiple will always be divisible by 4). Thus, for a, b, and 3a+23b the GCD will be 4.
So, statement 2 is sufficient.
(2)- Since the GCD of a, b, 23a+3b is 4, we can conclude that either a or b is equal to 4 and both are divisible by 4. Now, since a and b are divisible by 4, 3a+23b will also be divisible by 4(the product of a multiple of 4 and any other multiple will always be divisible by 4). Thus, for a, b, and 3a+23b the GCD will be 4.
So, statement 2 is sufficient.
