If n has 9 factors, how many factors does 6n have? (1) n is

Question

If n has 9 factors, how many factors does 6n have?

(1) n is divisible by 175.
(2) Both 2n and 3n have 18 factors.

2times · Answer

Ans  D 

This one is tricky....
For any given no we can calculate the total no of factors if we know its prime factors, and the no of times they repeat.
lets say the prime factors of n are x(repeated a times), y (repeated b times), z(repeated c times)

Now the no of total factors of any number is given as a product of 1+ the power of each prime number in the number n. ...(1)

So if n = 12 = 2*2*3 = 2^2 * 3^1 total no of factors = (2+1)*(1+1) = 3*2 =6

Now we also know that a number can have an odd no of factor only if it is a perfect square; which means in our number 'n' atleast one prime factor is repeated

so lets say n = X^2 * y^b *z^c ( b & c were assumed earlier)
now using the formula in 1.

(2+1) *(b+1) *(c+1) = 9
this imples that 'n' will be = x^2 y^2

now to determine x & y....

1) n divisible by 175 (5*5*7) this immediately tells us the 'missing factor' is 7

so n = 5*5*7*7 ; now look at the factors of 6 n
6 n = 2*3* 5^2*7^2
now using (1) the # of factors = (1+1)*(1+1)*(2+1)*(2+1) = 36; thus sufficient

2) this statement just confirms that n and 6 n do not have 2& 3 as common factors....
which means n = x^2* y^2 * 2*3
again here we don't need to know the factors such the exponents of the prime factors ; so ans = 36 - sufficient



kevincan is there a shorter method?

kevincan · Answer

Not that I know of. Excellent work!

2times · Answer

btw, your questions are brilliant!
They definitely take longer than 2 minutes to solve (atleast for me), but solving each question clears atleast 2 -3 concepts.

Please keep the tricky ones coming!

lan583 · Answer

i just wanna know why do you need to one more 7, the one that you said was the missing factor. is 5*5*7 enough.. what am i missing in here.

2times · Answer

as the no of factors is given by 9; 9 = 3*3
= (2+1)*(2+1)
and 175 = 5^2 * 7^1; so to meet the pattern above 7 also has to be raised to power 2.

hope that helps!

2times · Answer

(2+1) *(b+1) *(c+1) = 9 --> how do you get from this... 

 this is the rule for estimating the number of factors; there was another q in this forum regaing the same 

this imples that 'n' will be = x^2 y^2 --> to this? Please explain 

 as 9 = 3*3; 
and in the previous statement we said 3 * (something else) = 9; 
so that (something else) has to be another 3; 
and that (something else) is actually nothing but (1+ the power of the digit 7);

If n has 9 factors, how many factors does 6n have? (1) n is

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