If n and t are positive integers, what is the greatest prime factor of

If n and t are positive integers, what is the greatest prime factor of nt?

(1) The greatest common factor of n and t is 5 --> if n and t does not have any prime greater than 5 then the greatest prime factor of nt will be 5 (example: n=5 and t=5 or n=10 and t=15) BUT if n and/or t have some primes more than 5 then the greatest prime factor of nt will be more than 5 (example: n=35 and t=5 --> the greatest prime of nt is 7 or n=5 and t=55 --> the greatest prime of nt is 11)

(2) The least common multiple of n and t is 105 --> the least common multiple of two integers contains all common primes of these integers, thus 105 has all the primes which appear in both n and t --> the greatest prime factor of 105 is 7, hence it's the the greatest prime factor of nt (no greater factor can "appear" in nt if it's not in either of them). Sufficient.

Answer: B.

Hope it's clear.
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Statement 1: GCF(n,t) = 5

if n=15, t=10, greatest prime factor of nt = 5
if n=21, t=15, greatest prime factor of nt = 7. Hence insufficient.

Statement 2: LCM(n,t) = 105 = 3x5x7

7 is the greatest prime factor for 105.

either n or t or both must contain 7 as a factor. Therefore 7 is the greatest prime factor of nt. Sufficient.

Choose B.

Bunuel

It is quite agreed that LCM of two integers will contain all the primes of these integers, does the same hold true for GCF of two numbers as well?

The greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.

To find the GCF, you will need to do prime-factorization. Then, multiply the common factors (pick the lowest power of the common factors).

Hence the GCF will contain only the common primes of two number. For example, the GCF of 36=2^2*3^2 and 100=2^2*5^2 is 4=2^2.


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Hi All,

Prime factorization is a very strong tool while solving LCM-GCD questions. This question directly tests the process knowledge of prime factorization method of finding LCM-GCD.

For further refinement of your approach in LCM-GCD questions and avoiding fatal mistakes please go through the below article on mistakes committed by students in LCM-GCD questions.

3 Deadly Mistakes you must avoid in LCM-GCD questions

Hope this helps

Regards
Harsh
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Hi guys, I got quite confused after seeing this question in my practice test.

There are no sets of numbers for n and t in whichh the GCF and 5 AND the LCM is 105. Sentences 1 and 2 therefore appear to be inconsistent.

My question is - for data sufficiency questions, aren't the 2 sentences supposed to be consistent with each other and the initial question? Or am I missing something here?

Thanks!

Posted from my mobile device

On the GMAT, two data sufficiency statements always provide TRUE information and these statements NEVER contradict each other or the stem.

But the statements in the original questions does not contradict. Consider
n = 5 and t = 105
n = 15 and t = 35

The greatest common factor of n and t is 5 and the least common multiple of n and t is 105.
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I keep stumbling wrt statement 2 here. What if n is 11x7x5x3 and t is 7x5x3. Then the LCM is still 7x5x3=105 but the greatest prime factor of nt is 11?

You are mixing the least common multiple and the greatest common factor. 105 is the greatest common factor (GCF) of 11*7*5*3 and 7*5*3, not the least common multiple (LCM).
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Got it. I knew I was messing up something. Thanks.

Dear Bunuel,

In Statement 1:

If n = 5 & t =45....then GCF =5 but what is the greatest prime factor of nt?? Is it 3^2 or 5?? When used term greatest, should we include power or ignore it?

Thanks

3^2 = 9 and it's not prime, so you should not consider it.

If n = 5 and t = 45, then \(nt = 5*45 = 3^2*5^2\). So, the greatest common factor of nt is 5.
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[quote="enigma123"]If n and t are positive integers, what is the greatest prime factor of nt?

(1) The greatest common factor of n and t is 5
(2) The least common multiple of n and t is 105



1- it's possible that n = t = 5. then the greatest prime factor of nt is 5.
- it's possible that n = 5 and t = 35. then the greatest prime factor of nt is 7.
insufficient.

2- the least common multiple contains every factor of t or n at least once. (it has to; if, say, t had a factor that wasn't contained in it, then it would fail to be a multiple of t.) so, the biggest prime factor of this # will also be the biggest prime factor of the product nt.
sufficient.

LCM contains all the factor hence ,only B is sufficient

If n and t are positive integers, what is the greatest prime factor of nt?

(1) The greatest common factor of n and t is 5
(2) The least common multiple of n and t is 105

Solution:
(1) \(GCF = 5\) thus \(n=(5)(n)\) \(t=(5)(n2)\) we don't know if n=n2 NOT SUFFICIENT.
(2) \(LCM = 105 = 5*7*3\) --> Biggest prime factor= 7 Sufficient.

Answer B

Statement 1-The greatest common factor of n and t is 5

Think of two numbers like 5 and 65. If n=t=5,then the greatest prime factor of nt shall be 5.However,if n=5 and t=65,the greatest prime factor of nt i.e.5 * 13 * 5 shall be 13 and this is not equal to 5. Hence, we do not have a definite answer for this.
(insufficient)

Statement 2-The least common multiple of n and t is 105

We know that 105 = 3 x 5 x 7

We can observe that 7 is the greatest prime factor for 105 and hence the inference is either n or t or both shall have 7 as a prime factor and thus it is the greatest prime factor of nt.
(sufficient)
(option b)

Devmitra Sen
GMAT SME

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This is a tricky question for sure. I suggest using the following examples as n and t for each condition:

1) n =1 and t =5 (answer is 1), n =5 and t =5 (answer is 5), NOT SUFFICIENT.

2) n =5 and t =105, (answer is 7), n =105 and t =105 (answer is 7), n = 210 and t = 105 (answer is 7), etc., SUFFICIENT.

Yes, the values for nt will eventually become quite large, but they are essentially pointless to calculate by hand, because the building blocks aka prime factorials of each multiplied number are always the same. Thus, the greatest common prime factor (GCPF) of nt never increases.

Pro Tips for GCF and LCM:

The GCF (greatest common factor) will never be LARGER than the SMALLEST number.

THE LCM (least common multiple) will never be SMALLER than the LARGEST number.

Did this is in 30 seconds.

The greatest prime number in any of the number i.e. N or T will be the greatest prime factor.

I. GCF will not reveal what is the greatest, rather it will reveal what is the common greatest, hence insuff.
II. LCM takes all the prime number in consideration, hence we can tell what is the greatest prime factor. sufficient.
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