Made Easy Booklet

Question

How many factors of
N=12^12×14^14×15^15
are multiple of
K=12^10×14^10×15^10 ?

Answer 1

 

 N=$12^{12}×14^{14}×15^{15}$

 

doing prime factorization of base:

N=$(2^2 ×3)^{12} × (7×2)^{14} ×(5×3)^{15}$

N=$2^{24} ×3^{12} × 7^{14}×2^{14} ×5^{15}×3^{15}$

N=$2^{38} ×3^{27} × 7^{14} ×5^{15}$

 

and K= $12^{10}×14^{10}×15^{10}$

doing prime factorization of base:

K= $(2^2 ×3)^{10} × (7×2)^{10} ×(5×3)^{10}$

K= $2^{20} ×3^{10} × 7^{10}×2^{10} ×5^{10}×3^{10}$

K= $2^{30} ×3^{20} × 7^{10}×5^{10}$

 

now we can see

N=$2^{38} ×3^{27} × 7^{14} ×5^{15}$

N=$(2^{30} ×3^{20} × 7^{10}×5^{10}) × (2^{8} ×3^{7} × 7^{4} ×5^{5})$

N=$ K× (2^{8} ×3^{7} × 7^{4} ×5^{5})$

 

concept:

Finding the Number of Factors

We can find the number of factors of a given number using the following steps.

• Step 1: Do the prime factorization of the given number, i.e., express it as the product of primes.
• Step 3: Write the prime factorization in the exponent form.
• Step 3: Add 1 to each of the exponents.
• Step 4: Multiply all the resultant numbers. This product gives the number of factors of the given number.

 

So the number of factors of N that are a multiple of K are

(8+1) × (7+1) × (4+1) × (5+1)

=9×8×5×6

=2160.