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kiran120680
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06 Jun 2019, 09:30
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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a and b are positive integers such that they do not have any common prime factor. What is the remainder when the positive integer c is divided by the lowest number that has both a and b as its factors?
(1) c has the same number of factors as that of the least common multiple of a and b.
(2) When c is divided by 4 times the product of a and b, the remainder is 0
(1) c has the same number of factors as that of the least common multiple of a and b.
(2) When c is divided by 4 times the product of a and b, the remainder is 0
09 Jun 2019, 07:28
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aliakberza
If a number is divisible by d, say, then the number will also always be divisible by every factor of d. So, for example, if a number is divisible by 12, it must also be divisible by 6, 4, 3 and 2. You can see why by factoring - if a number is divisible by 12, then it equals 12q, where q is a whole number. But then it also equals 6(2q), so it's a multiple of 6 too, for example, and it also equals 4(3q), so it's a multiple of 4, and so on.
So if a number is divisible by 4ab, it is divisible by every divisor of 4ab, so it's divisible by ab, and by 4, and by 4a, and by 2b, and by several other things.
Notice in your example, where a = 3 and b = 5, then ab = 15, and 4ab = 60. That number is not divisible by 16, since 60/16 is not an integer -- it does share some divisors with 16, so you can cancel that fraction down a bit, to 15/4, but you don't get an integer when you're finished canceling.
General Discussion
06 Jun 2019, 10:17
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We want the remainder when c is divided by the LCM of a and b. But if a and b share no prime divisors, the LCM of a and b is just equal to ab (though that doesn't turn out to be important here - the answer would still be B even if that were not true, since the LCM of a and b is a factor of ab anyway).
I'm not sure what the "that of" part of Statement 1 even means, but if I assume they're saying that c and ab have the same number of divisors, then Statement 1 is useless. Lots of numbers have the same number of divisors, and you can't from that fact conclude anything about remainders when you divide one of those numbers by the other. Maybe c = ab, and the remainder is zero, or maybe c and ab are different, and the remainder is nonzero.
Statement 2 tells us c is divisible by 4ab, so c must be divisible by ab, so Statement 2 is sufficient; the remainder is zero.
I'm not sure what the "that of" part of Statement 1 even means, but if I assume they're saying that c and ab have the same number of divisors, then Statement 1 is useless. Lots of numbers have the same number of divisors, and you can't from that fact conclude anything about remainders when you divide one of those numbers by the other. Maybe c = ab, and the remainder is zero, or maybe c and ab are different, and the remainder is nonzero.
Statement 2 tells us c is divisible by 4ab, so c must be divisible by ab, so Statement 2 is sufficient; the remainder is zero.
