In how many ways the number 7056 can be written as a product of 2 posi

Question

In how many ways the number 7056 can be written as a product of 2 positive factors?

A. 20
B. 21
C. 22
D. 23
E. 24

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CrackverbalGMAT · Accepted Answer

We have that for any number, N broken into prime factors and their powers in the form  N = a^p * b^q*c^r  and so on, then the total number of factors n = (p + 1) * (q + 1) * (r + 1) and so on.

If the number of factors is even, then the number of pairs that give the product as N =  \frac{n}{2}

If the number of factors is odd, then the number of pairs that give the product as N =  \frac{n + 1}{2}

N = 7056 =  2^4 * 3^2 * 7^2

n = (4 + 1) * (2 + 1) * (2 + 1) = 5 * 3 * 3 = 45

Therefore the number of pairs =  \frac{45 + 1}{2} = 23

Option D

Arun Kumar

ShaikhMoice · Answer

Could you please elaborate the coloured part a little bit.
Thanks in Advance

CrackverbalGMAT · Answer

If we write down any number N in their prime factorization form as N =  x^{a} *{y^b}*{z^c } and so on.
Where x,y,z are prime numbers.
Then the total no of factors of N =(a+1)(b+1)(c+1) and so on.
Let’s try it with an example, N = 20

We can prime factorize N as    {2^2 } *{5^1}

We can use the above mentioned formulae to find the number of factors =   (2+1)*(1+1) =3*2= 6  
Let’s confirm it:
The factors of 20 are1,2,4,5,10,20
So, we have a total of 6 factors

The pairs of factors that give the product N are  (1,20), (2,10) ,(4,5) 
So total no of pairs we got are 3.
Hence, we can generalise it as,  if the no of factors is even, then the no of pairs of factors that give the product as N=  (no of factors)/2  
In the above example it would be 6/2 = 3.

Take another number N = 16

We can prime factorize N as   2^4

No of factors = (4+1) = 5
Let’s write down the factors of 16
They are 1,2,4,8,16

So pairs of factors that give the product N are   (1,16), (2,8) ,(4,4) 
The total no of pairs is only 3.
We can generalise it as,  if the no of factors is odd, then the no of pairs of factors that give the product as N=  (no of factors +1)/2  = (5+1)/2 = 3

Hope this explanation helps.

Thanks,
Clifin J Francis

MathRevolution · Answer

7056 = 2^4 * 3^2 * 7^2

Total factors: (4 + 1)(2 + 1)(2 + 1) = 5 * 3 * 3 = 45

We need two positive factors:  \frac{45 + 1 }{ 2} = 23  (Since the number of total factors are odd)

Answer D

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In how many ways the number 7056 can be written as a product of 2 posi

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