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Re: GMAT CLUB OLYMPICS: What is the number of factors of a positive intege
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20 Aug 2021, 12:31
The Lowest Common Multiple (LCM) of two or more numbers is the smallest of the common multiples of those numbers. Thus LCM multiplies smallest common multiples which are present in both/more numbers, as well as non-common multiples that is/are exclusive to one of the numbers.
The common multiples are multiplied once for each occurrence in common, or multiplied once for one exclusive occurrence to one of the number.
Say, for example, to find LCM of 20 and 42,
first number = 2,2,5
second number = 2,3,7
Here the LCM is 2x2x3x5x7 = 420.(demonstration purpose only, to show how to calculate LCM)
(1) LCM of n/2 and n/3 is 210
In (1) dividing n by 2 and 3 and using those results are used to find the LCM.
Multiples of 210 = 2,3,5,7
Since the number 210 already contains 2 and 3, we can easily say that n/2 and n/3 has numbers 5 and 7 present as common multiples while 2 is present only in n/3 and 3 was present only in n/2 => we can prove that 2 and 3 multiple occured only once.
So, we can say that the number n had all the multiples that occurred only once, and is 2x3x5x7 = 210. (1) alone is sufficient. --------(i)
Looking at option (2),
Greatest common factors of two or more given numbers is the largest number which divides each of the given numbers without leaving any remainder.
This means from the given two numbers n/5 and n/7 we removed one of the common multiple 5 and 7 respectively and found the largest number that was common is 6 (2,3). So there is only one occurence of 2 and 3 in the number. Because we also can prove n/5 eliminated only one 5 and n/7 eliminated only one 7, then we can say that the multiple appeared only once and they are 2,3,5 and 7. So the actual number n must be 2x3x5x7 = 210.
so, (2) alone is sufficient. ----(ii)
Based on (i) and (ii),
The right choice is D.