Question

What is the highest power of 24 th:
an divide 70!​

Answer 1

Answer:

t is sufficient to find the power of 5 in the prime factoring of 70! (the power of 2 is obviously larger than the power of 5)

This is ⌊70/5⌋+⌊70/25⌋=14+2=16. The answer is 1016

Answer 2

SOLUTION

TO DETERMINE

The highest power of 24 in 70!

FORMULA TO BE IMPLEMENTED

1. The highest exponent of the prime number p in n! is

$= \displaystyle \sf{\sum\limits_{k=1}^{m} \: \: \frac{n}{ {p}^{k} } \: }$

$\sf{\: where \: k \: is \: the \: smallest \: integer \: such \: that \: \: {p}^{k +1} > n}$

2. Box Function :

$\sf{ \: [ x ] = the \: greatest \: integer \: but \: not \: greater \: than \: x }$

CALCULATION

Here

$24 = 2 \times 2 \times 2 \times 3 = {2}^{3} \times 3$

So in the prime factorisation of 24 only two prime numbers 2 & 3 are present

So we will proceed to find the power of 2 & 3 in 70!

Now

$\sf{ {2}^{7} > 70 }$

So the largest power of the prime number 2 in 70! is

$\displaystyle \: \sf{ = \bigg[\: \frac{70}{2} \bigg]} + \: \sf{\bigg[\: \frac{70}{ {2}^{2} } \bigg]} + \: \sf{\bigg[\: \frac{70}{ {2}^{3} } \bigg]} + \: \sf{\bigg[\: \frac{70}{ {2}^{4} } \bigg] + \sf{\bigg[\: \frac{70}{ {2}^{5} } \bigg]} +\sf{\bigg[\: \frac{70}{ {2}^{6} } \bigg]} }$

$\displaystyle \: \sf{ = \bigg[\: \frac{70}{2} \bigg]} + \: \sf{\bigg[\: \frac{70}{ 4} \bigg]} + \: \sf{\bigg[\: \frac{70}{ 8 } \bigg]} + \: \sf{\bigg[\: \frac{70}{ 16 } \bigg] + \sf{\bigg[\: \frac{70}{ 32 } \bigg]} +\sf{\bigg[\: \frac{70}{ 64 } \bigg]} }$

$\displaystyle \: \sf{ =35 + \: 17 + \: 8 + 4 + 2 +1 }$

$\displaystyle \: \sf{ =67 }$

Again

$\sf{ {3}^{4} } > 70$

So the largest power of the prime number 3 in 70! is

$\displaystyle \: \sf{ = \bigg[\: \frac{70}{3} \bigg]} + \: \sf{\bigg[\: \frac{70}{ {3}^{2} } \bigg] +\sf{\bigg[\: \frac{70}{ {3}^{3} } \bigg]} }$

$\displaystyle \: \sf{ = \bigg[\: \frac{70}{3} \bigg]} + \: \sf{\bigg[\: \frac{70}{ 9 } \bigg] +\sf{\bigg[\: \frac{70}{ 27} \bigg]} }$

$\displaystyle \: \sf{ = 23 + 7 + 2 }$

$\displaystyle \: \sf{ =32 }$

Hence the highest exponent of 24 in 70! is

$\displaystyle \: \sf{ = min \bigg \{ \bigg[\: \frac{67}{ 3 } \bigg] \: ,\bigg[\: \frac{32}{ 1 } \bigg] \bigg \} }$

$\displaystyle \: \sf{ = min \bigg \{ 22 \: ,32 \bigg \} }$

$\displaystyle \: \sf{ =22 }$

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