A number theory problem by Akshat Sharda

Number Theory Level 4

P P is the product of all the factors of 15552 15552 .If P = 1 2 n × m q P=12^{n}×m^{q} where m m is not a multiple of 12 12 .[ n , m n,m and q q are positive integers].

Find n + m + q n+m+q .


The answer is 108.

Akshat Sharda by Akshat Sharda , 21, India

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1 solution

Akshat Sharda
Jul 25, 2015

15552 = 2 6 × 3 5 15552 = 2^{6}×3^{5}

Now we have a formula for product of all the factors - N N u m b e r o f f a c t o r s 2 N^{\frac{Number of factors}{2}}

Product of all factors of 15552 15552

= ( 2 6 × 3 5 ) ( 6 + 1 ) ( 5 + 1 ) 2 (2^{6}×3^{5})^{\frac{(6+1)(5+1)}{2}}

= 2 126 × 3 105 2^{126}×3^{105}

= ( 2 2 ) 63 × 3 63 × 3 42 (2^{2})^{63}×3^{63}×3^{42}

= 1 2 63 × 3 42 12^{63}×3^{42}

Therefore,

n + m + q = 63 + 3 + 42 = 108 n+m+q = 63+3+42 = \color{#D61F06}{\boxed{108}}

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Akshat Sharda - 5 years, 10 months ago

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