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# If n = 2pq, where p and q are distinct prime numbers greater

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Re: If n = 2pq, where p and q are distinct prime numbers greater [#permalink]
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megafan
If n = 2pq, where p and q are distinct prime numbers greater than 2, how many different positive even divisors does n have, including n ?

(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight

Since we cannot have two correct answers just pick two primes greater than 2, and see how many different positive even divisors will 2pq have.

Say p = 3 and q = 5 --> 2pq = 30--> 30 has 4 even divisors: 2, 6, 10 and 30.

Similar question to practice from OG: if-n-2pq-where-p-and-q-are-distinct-prime-numbers-greater-148939.html

Another similar question: if-n-is-a-prime-number-greater-than-3-what-is-the-remainder-137869.html

Hope it helps.

Total number of divisor of 2pq are (1+1)*(1+1)*(1+1)= 8
Total number of odd divisor of 2pq are (1+1)*(1+1)=4
Total number of even divisor of 2pq are = total divisor- odd divisor=8-4=4
Hence c
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Re: If n = 2pq, where p and q are distinct prime numbers greater [#permalink]
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megafan
If n = 2pq, where p and q are distinct prime numbers greater than 2, how many different positive even divisors does n have, including n ?

(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight

Because p and q are distinct prime number > 2, so p and q are ODD numbers

2pq = 2*odd*odd

There are only four EVEN factors of n, including n (because n is even):
2
2*p
2*q
2*pq

C is correct.

Hope it helps.
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Re: If n = 2pq, where p and q are distinct prime numbers greater [#permalink]
Divisors are:

n, 2, 2p, 2q

C
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Re: If n = 2pq, where p and q are distinct prime numbers greater [#permalink]
Even integer multiplied by any number whether odd/even will ALWAYS give even integer.

For example, 2*3*3*3*3*3*3= 1458

Therefore in the above question there will be 4 even positive divisors of n.
n=2*p*q
p,q>2

2 is the only even prime number.
Thus p and q are odd and will be counted under odd divisors. p*q (odd*odd) will also give odd.

Thus the even divisors are: 2, 2p, 2q, and 2pq
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Re: If n = 2pq, where p and q are distinct prime numbers greater [#permalink]
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megafan
If n = 2pq, where p and q are distinct prime numbers greater than 2, how many different positive even divisors does n have, including n ?

(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight

We can let p = 3 and q = 5. Thus, the product of 2pq is 2 x 3 x 5 = 30. The factors of 30 are:

1, 30, 2, 15, 3, 10, 5, 6

Since 30 has 4 even factors, n has 4 even factors.

Alternatively, we can solve the problem algebraically. Keep in mind that p and q will be odd primes since they are greater than 2.

The factors of n are:

1, 2pq, 2, pq, p, 2q, q, 2p

We see that the even factors of n are 2pq, 2, 2q, and 2p, so there are 4 even factors.

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If n = 2pq, where p and q are distinct prime numbers greater [#permalink]
If n = 2pq, where p and q are distinct prime numbers greater than 2

Pick any two prime numbers greater than prime 2. Say 5 and 7

n = 2 x 5 x 7

Factor of n is n itself - 70

and the even factors of n are 5 x 2 = 10

7 x 2 = 14 and 2

Therefore, total factors of n are 2,10, 14 and 70 - 4 (C)

Originally posted by medinib on 15 Jan 2020, 09:34.
Last edited by medinib on 15 Jan 2020, 11:46, edited 1 time in total.
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Re: If n = 2pq, where p and q are distinct prime numbers greater [#permalink]
megafan
If n = 2pq, where p and q are distinct prime numbers greater than 2, how many different positive even divisors does n have, including n ?

(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight

n=2pq
Since p & q are odd prime numbers greater than 2, p & q are not even
2, 2p , 2q and 2pq are even divisors.
4 numbers

IMO C

Posted from my mobile device
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Re: If n = 2pq, where p and q are distinct prime numbers greater [#permalink]
If n = 2pq, where p and q are distinct prime numbers greater than 2, how many different positive even divisors does n have, including n ?

Ans:
1. P& q are odd, so pq is odd but when multiplied by 2, it will yield even number, so n is even
2. 2 is also a factor of n
3 2p is also going to be an even factor
3 2q is also going to be an even factor

so answer is four is (C)
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Re: If n = 2pq, where p and q are distinct prime numbers greater [#permalink]
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Re: If n = 2pq, where p and q are distinct prime numbers greater [#permalink]
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