If a and b are consecutive positive integers, and ab = 30x

Question

If a and b are consecutive positive integers, and ab = 30x, is x an integer?
 
(1) a^2 is divisible by 25
(2) 63 is a factor of b^2

Bunuel · Accepted Answer

If a and b are consecutive positive integers, and ab = 30x, is x an integer?

ab = 30x  -->  x=\frac{ab}{30} . x to be an integer ab must be a multiple of 30.

(1) a^2 is divisible by 25. This implies that a is a multiple of 5. If a=5 and b=4, then ab=20 and x is NOT an integer but if a=5 and b=6, then ab=30 and x IS an integer. Not sufficient.

(2) 63 is a factor of b^2 -->  63=3^2*7 . Thus b is a multiple of 3 (and 7). If b=21 and a=22, then ab=21*22 and x is NOT an integer but if b=21*30, then ab={a multiple of 30} and x IS and integer. Not sufficient.

(1)+(2) We have that a is a multiple of 5 and b is a multiple of 3. Also, since they are consecutive integers, then either one must be even. Therefore ab is a multiple of 2*3*5=30. Sufficient.

Answer: C.

Hope it's clear.

doan8606 · Answer

In order for ab = 30x = 2*3*5*x, ab needs to be divisible by 2,3 and 5 (because x can be any positive number from 1, 2...)
 -a and b are consecutive positive integer, so one must be even and the other is odd, so ab will be divisible by 2.
 -a^2 is divisible by 25 = 5^2, so a is divisible by 5.
 -63 = 3^2 * 7 is a factor of b^2, so b is divisible by 3 and 7.
As you can see, the condition a alone is not sufficient because we only know ab will be divisible by 2 and 5. Condition b alone is not sufficient also because we only know ab will be divisible by 2, 3 and 7. If a and b together, we know ab = 2*5*3*7*something, so sufficient.
Therefore, the correct answer is C.

Hope this help!

mpingo · Answer

I have one doubt. If b^2 has a factor 63
Then, b^2 has factors 3, 3, 7 for sure atleast once.
Then, how can we be so sure that b has also 3 and 7 as factors. isnt b has only 3 and \sqrt{7} as factors?

chetan2u · Answer

Hi,
b is a positive integer so it cannot be3* \sqrt{7} * something else... 
therefore it has to be a multiple of 7 and not  \sqrt{7}

Bunuel · Answer

1. Only positive integers are considered as factors. So, saying that \sqrt{7} is a factor of some integer does not make sense. 2. Exponentiation does not "produce" primes, meaning that for positive integers m and n, m^n will have the same exact prime factors as m itself does. Thus, if 3 and 7 are factors of b^2 it must be true that both of them are factors of b too (else how would they appear in b^2). Similar questions to practice: if-x-is-an-integer-is-x-3-divisible-by-165973.html how-many-different-prime-numbers-are-factors-of-the-positive-126744.html if-n-2-n-yields-an-integer-greater-than-0-is-n-divisible-by-126648.html if-k-is-a-positive-integer-is-k-the-square-of-an-integer-55987.html if-k-is-a-positive-integer-how-many-different-prime-numbers-95585.html if-n-is-the-integer-whether-30-is-a-factor-of-n-126572.html if-x-is-an-integer-is-x-2-1-x-5-an-even-number-104275.html Hope it helps.

energetics · Answer

Edit: reading above my solution is definitely not the best ... anyway some takeaways:
It's not ideal to assume b=a+1, it could be b=a-1 (doesn't matter here since it just means one is E, other is O so divisible by 2 but it could be an issue in another problem)
Also, even not knowing what bunuel stated about prime factor 7 being in b, it is irrelevant because we're only looking for divisibility by 3,5

a and b are consecutive positive integers means b = a+1 
a(a+1) = 30x, 
a(a+1)/30 = x
Question: is a(a+1) / 3*2*5?

(1) a^2 is divisible by 25 
Try values: a=25, a+1 = 26
(5*5) * (2*13 ) / 3*2*5? 
No, there isn't a factor of 3 in numerator

a = 125, a+1 = 126
(5*5*5) * (2*3*3*7) / 3*2*5? 
Yes, enough factors of everything

Not sufficient.

(2) 63 is a factor of b^2 
Try values: a+1 = 63, a= 62
(3*3*7) * (2*31) / 3*2*5? 
No, there isn't a factor of 5 in numerator

(a+1) = 126 , a = 125
From 1) we know this gives a Yes.

Not sufficient.

(3) 
The same example satisfies both 1) and 2), sufficient.

bumpbot · Answer

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If a and b are consecutive positive integers, and ab = 30x

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