If q and r are prime numbers and n = 4qr, which of the following could

Question

If q and r are prime numbers and n = 4qr, which of the following could be the number of factors of n that are between 1 and n, exclusive ?

(I) 3
(II) 4
(III) 7

(A) I only 
(B) II only 
(C) I and III only 
(D) II and III only 
(E) I, II and III

suchita2409 · Answer

IMO E.

We know that p and r are prime factors.
 4 is not prime , so 2^2 so it has (2+1)=3 factors and p and r has (1+1) 2 factors each factors and therefore n has 3*2*2
= 12 factors. 
It's a could-be question, so all 3 can be no. of factors. Therefore E.

harvardforsure · Answer

If q and r are distinct and none of them is 2 
then no of factors is 3*2*2=12. Less 2 gives you 10 factors

If one of them is 2 
Then no of factors is 4*2-2=6 factors

If both of them are 2 (not given that they are distinct)
Then factors if 4+1=5 -2 = 3 factors

wali786 · Answer

The answer is E, because there are 12, 9 or 5 factors and 3 and 7 can be in of those factors.

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TarunKumar1234 · Answer

If q and r are prime numbers and n = 4qr, which of the following could be the number of factors of n that are between 1 and n, exclusive ?

(I) 3-> if q=r=2 then n= 4*2*2 = 2^4, no. of factors between 1 and n= (4+1) - 2 = 3. correct.
(II) 4->if q=2, r=3 or 5 then n= 4*2*3 = 2^3 * 3^1, no. of factors between 1 and n= (3+1)*(1+1) -2 = 6. Incorrect.
(III) 7->if q=r=3 or 5 then n= 4*3*3 = 2^2 * 3^2, no. of factors between 1 and n= (2+1)*(2+1) -2 = 7. correct.

So, I think C.

TH17 · Answer

n = 4qr => n = (2^2)*q*r
there are 4 cases here:
case 1: if q=r=2
n = 2^4
number of factors excluding 1 and n = (4+1) - 2 = 3

case 2: if q=r but not equal to 2 ( lets say q=r=k )
n = (2^2)*(k^2)
number of factors excluding 1 and n = (2+1)(2+1) - 2 = 7

case 3: if q and r are not equal and each not equal to 2
n = (2^2)*q*r
number of factors excluding 1 and n = (2+1)(1+1)(1+1) - 2 = 10

case 4: if q and r are not equal and but one of them is equal to 2 ( lets assume q=2 )
n = (2^3)*r
number of factors excluding 1 and n = (3+1)(1+1) - 2 = 6

Hence C

EgmatQuantExpert · Answer

Given

• q and p are prime numbers.
• n = 4qr

To Find

• The number of factors of n between 1 and n, exclusive.

Approach and Working Out

• There are multiple scenarios possible.

o Scenario 1: p = q = r
o Scenario 2: p = 2, q = 3 or any other prime number apart from 2. (or vice versa)
o Scenario 3: p = q = 3 or any other prime number apart from 2.
o Scenario 4: p = 3 and q = 5 (any other combo where p and r are different prime numbers and none of them is 2)

• The number of factors for these numbers will be,

o Scenario 1: n = 16, factors = 5, number of factors other than 1 and n = 3.
o Scenario 2: n = 24, factors = 8, number of factors other than 1 and n = 6.
o Scenario 3: n = 36, factors = 9, number of factors other than 1 and n = 7.
o Scenario 4: n = 60, factors = 12, number of factors other than 1 and n = 10.

• We can say only I and III are correct.

Correct Answer: Option C

MHIKER · Answer

p  and  r  are can be equal and  p  and  r  can be different.

p=r=1   n=4pr=4;  Total factors =3;  Quesiton asked exclusive. So, the factors  3-2=1 , Not True

p=1; r=2; n=4pr=4*1*3=12;  Total factors =6 , Quesiton asked exclusive. So, the factors  6-2=4  Not True

p=2; r=2; n=4pr=4*2*2=16;  Total factors=5, Quesiton asked exclusive. So, the factors  5-2=3; \ True

p=3; r=3; n=4pr=4*3*3=36 ; Total factors =9 , Quesiton asked exclusive. So, the factors  9-2=7; \ True

The answer is  C

rocky620 · Answer

Since it is exclusive of 1 & n we have to subtract 2 factors
 Case-1:  q=r=2; n=4*2*2 =  2^4 . No of factors = 4+1 = 5, Subtract 2 =  3 
 Case-2:  q=r=p (which is any prime other than 2); n=4* p^2 ; =  2^2 * p^2 ; no of factors = (2+1)*(2+1) = 9, subtract 2 =  7 
 Case-3: q & r are different prime numbers, other than 2 (such as 3&5); n=  2^2 *q*r; no of factors = (2+1)*(1+1)*1+1) = 12, subtract 2 =  10 
 Case-4: q or r is 2; n=  2^3 *r; no of factors = (3+1)*(1+1) = 8, subtract 2 =  6

No other case is possible, so only 3 & 7 are possible, Option C.

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If q and r are prime numbers and n = 4qr, which of the following could

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If q and r are prime numbers and n = 4qr, which of the following could

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