factors

[Jaguar36608]

If n=2^4*3*5, how many factors of n are greater than or equal to 8 and less than or equal to 30?
(A) 8
(B) 9
(C) 10
(D) 12
(E) 15

Please explain your method. It took me around 4 mins to solve this question.

Thank you.

[bikerguy.baj]

i got the answer as 8 , but not very convinced with my approach .
listed out all the factors of n and try to circle out the ones which are between 8 and 30 . took about 2 mins to do this .

i too would like to know the best way of doing this .

[Jaguar36608]

Anyone else wants to give a shot?

[GmatG]

there is a formula to calculate the total no. of factors of n :

n = x^a * y^b * z^c .. where x,yz are prime no. and a,b,c are +ve integers..
than, total no. of factors of n will be
(a+1)(b+1)(c+1).

Applying the above formula in the question.. n = 2^4 * 3 * 5

a=4, b=1 and c = 1

So, total no. of factors will be (4+1)(1+1)(1+1) ie, 20. --- (1)

factors of n are greater than or equal to 8 and less than or equal to 30factors of n are greater than or equal to 8 and less than or equal to 30

8<= factor <= 30
2^3 <= factor <= 2*3*5

now, finding no. factors less than 2^3
(a =0, b=0 and c=0) 1
(a =1, b=0 and c=0) 2
(a =2, b=0 and c=0) 4
(a =0, b=1 and c=0) 3
(a =0, b=0 and c=1) 5
(a =1, b=1 and c=0) 6

no. of fators less than 2^3 will be 6

no. of fators greater than 2*3*5
(a =2, b=1 and c=1) 60
(a =3, b=1 and c=1) 120
(a =4, b=1 and c=1) 240
(a =4, b=1 and c=0) 48
(a =4, b=0 and c=1) 80
(a =3, b=0 and c=1) 40
will be 6

Total no. of factors that does not fall into the range of
2^3 <= factor <= 2*3*5 will be 12 (6+6) ---- (2)

hence required no. of factors = 8 (1) - (2)

Hope that helps. Thanks

[Retake_GMAT]

Great Solution GmatG!!

[raunekk]

excellent method GmatG....

But is there a shorter method for this???

[sanjeev_sharma]

Anyone have any shortcuts for solving this kind of a problem