SOLUTION

Does the integer k have at least three different positive prime factors?

(1) k/15 is an integer --> since k is divisible by 3*5=15, then it's divisible by 3 and 5, hence k has at least those two prime factors, though it might have more (consider k=30). Not sufficient.

(2) k/10 is an integer --> since k is divisible by 2*5=10, then it's divisible by 2 and 5, hence k has at least those two prime factors, though it might have more (consider k=30). Not sufficient.

(1)+(2) At least 3 primes are factors of k: 2, 3, and 5. Sufficient.

Answer: C.
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Hi,

For integer k to have atleast 3 different positive prime factors,
k = a.b.c, where a, b, c are prime numbers.

Using (1),
k/15 is an integer,
least value of k =15, which has two different prime factors 3 & 5. Insufficient.

Using (2),
k/10 is an integer,
least value of k =10 which has two different prime factors 2 & 5. Insufficient.

Using both,
least value of k =30 which has two different prime factors 2, 3 & 5. Sufficient.

Answer is (C)

Regards,

If a divides b, then all the primes in a along with their powers are in b
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0 is a multiple of all positive integers. Thus if k=0 it still has at least three different positive prime factors.
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Here is my solution->
We need to check if the number of prime factors of p are atleast 3 i.e.≥3
Statement 1
p=15 => no
p=15*17 => yes
Not sufficient
Statement 2
p=10 => no
p=10*13 => yes
Not sufficient
Combining the two statements we can say that p=2*3*5*x for some integer x.
Clearly p must have atleast 3 prime factors
Hence C

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I solved this problem using Plug In

1, If k=30, have 2,3,5 prime factors
If K=75, have 3,5 only
Insufficient
2, If k = 70 , have 2,5,7 prime factors
If k = 80, Only 2,5 prime factors
Insufficient
3, K/15 and K/10
If k=30 2,3,5
K=90 2,3,5 prime factors

Irrespective K value , these only both meet conditions
C

Is my approach Right?
Thanks

Here is my solution:

Question: Does K have at-least 3 different positive prime factors.

Statement 1: \(\frac{k}{15}\) is an integer.

Hence the values of k are multiples of 15 such as 15,30,45...
Factors of 15 = 5 and 3 (2 factors)
Factors of 30 = 5 , 3 and 2 (3 factors)
Hence not sufficient.

Statement 2: \(\frac{k}{10}\) is an integer

Hence the values of k are multiples of 10 such as 10,20,30
Factors of 10 = 5 and 2 (2 factors)
Factors of 30 = 5 , 3 and 2 (3 factors)
Hence not sufficient.

Stmt 1 + Stmt 2:

k should be multiples of 10 and 15 such as 30,60...
All these values have atleast 3 positive different prime factors.

Hence C.
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Hi, does 1 counted as a prime factor?

No, 1 is not a prime number. The smallest prime is 2.
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We need to determine whether k has at least three different prime factors.

Statement One Alone:

k/15 is an integer.

Statement one alone is not sufficient to answer the question. If k = 15, then k has two different prime factors; however, if k = 30, then k has three different prime factors.

Statement Two Alone:

k/10 is an integer.

Statement two alone is not sufficient to answer the question. If k = 10, then k has two different prime factors; however, if k = 30, then k has three different prime factors.

Statements One and Two Together:

Using statements one and two, we see that k is a multiple of both 10 and 15, and thus it is a multiple of their least common multiple, which is 30. Since all multiples of 30 have at least three different prime factors, the two statements together are sufficient.

Answer: C
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This concept is discussed in detail here:
https://www.veritasprep.com/blog/2014/0 ... r-factors/
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Could integer k be 0?

If so, both conditions (k/10 is an integer) and (k/15 is an integer) would both be met as 0/anything = 0. Does 0 have an infinite number of prime factors?

Thank you all!

Hello,

Bunnel has already mentioned this in above thread..


0 is a multiple of all integers except 0 itself. Thus if k=0 it still has at least three different positive prime factors.

Hope it helps !!

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