III to be true it should be mentioned that k is a positive integer. For example, 0 is a common multiple of 75, 98, and 140 and is NOT greater than 14,000.

Rare example of flawed question from GMAT.
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Prime factors:-
\(75= 3*5*5\)
\(98= 2*7*7\)
\(140= 2*2*5*7\)
\(LCM= K= 2*2*3*5*5*7*7= 14700\)
I) K cannot be divisible by 9 since only a 3 is the factor of 14700
II) K is divisible by 49 as is evident by the two 7s
III) K>14000

So only (II) and (III) are always true hence the answer D

Given K is a common multiple of 75,98,140, Least Common Multiple of these numbers will be 14700
K will be of the form 14700*n...where n=1,2,3,.....

I)14700*n is not divisible by 9 for all values of 'n' (eg: at n=1,k=14700 not divisible by 9). Hence I cant be always true
II) 14700*n is divisible by 49 for all values of 'n'. Hence II is true
III)14,700*n is greater than 14,000.Hence III is true

Ans d) II and III only
Hope its clear

Prime factorization:

75=5*5*3
98=2*7*7
140=5*2*2*7

1) k is divisible by 9: NO, there is no prime factors of 9, i.e 3*3
2) k is divisible by 49: YES, there is 7*7=49
3) k>14,000: LCM=49*25*4*3=49*300=50*300>14,000, so YES

D

Bunuel whats your take?

I was also astonished it was an exam pack questions.

Bunuel, since its asking which of the following are true and NOT 'must be true'. This question seems legit, don't you think ?

Because i have seen official questions always mention 'must be true' if they are asking for only must be true answers, for example: Refer OG 13- PS- 221, but you are the Gmat Guru so please correct me if I'm wrong.

Thanks.

The question asks "which of the following statements are true?", which is equivalent to "which of the following statements must be true?". If it were otherwise it would be "which of the following statements could be true?"
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Bunuel, so it is either 'could be true' or 'must be true' and not as this question states. Got it.

Thanks.

Swaroopdev I'm sorry, but I don't understand what the difference of meaning is when "must" is left out. I suppose without "must" there is less emphasis, but other than that, the meaning is the same.

Bunuel, Calculating LCM of 75,98 and 140 is 14700 which is least,so how can you say zero is a common multiple? Please correct me if I am wrong.

Notice that we are told that k is a common multiple of 75, 98, and 140 NOT the least common multiple of 75, 98, and 140. By default, the least common multiple is considered to be the least common positive multiple, because otherwise it does not make any sense.

For example, the LCM of 2 and 3 is 6, which means that 6 is the least common positive multiple of 2 and 3. If we were allowed to consider negative multiples too, then there won't be an answer.

Back to the question:
1. 0 is a multiple of every integer, so 0 is a common multiple of every non-zero sets of integers.
2. 0 is not the only common multiple of 75, 98, and 140, less than 14,000, so is -14,700, -2*14,700, -3*14,700, -4*14,700, ...

Hope it's clear.
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