Find, with proof, the largest natural number k such that 10^k divides 100! (one hundred factorial).
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I was able to get to a theorem saying "that a|b if and only if for any prime in the factorization of a or b, its exponent in the factorization of a is less than or equal to its exponent in the factorization of b."
I tried to use this theorem where k <= m, k <= n, where m and n are exponents of 2 and 5, respectively, in the factorization of 100! but I was not able to work out the math so far.
Any help will be greatly appreciated!
divisibility
edited Nov 28 '18 at 21:56
amWhy
1
asked Nov 28 '18 at 21:51
UMass1234UMass1234
112
$endgroup$
add a comment |
$begingroup$
I was able to get to a theorem saying "that a|b if and only if for any prime in the factorization of a or b, its exponent in the factorization of a is less than or equal to its exponent in the factorization of b."
I tried to use this theorem where k <= m, k <= n, where m and n are exponents of 2 and 5, respectively, in the factorization of 100! but I was not able to work out the math so far.
Any help will be greatly appreciated!
divisibility
edited Nov 28 '18 at 21:56
amWhy
1
asked Nov 28 '18 at 21:51
UMass1234UMass1234
112
$endgroup$
1
$begingroup$
Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
$endgroup$
– Ross Millikan
Nov 28 '18 at 21:59
add a comment |
$begingroup$
I was able to get to a theorem saying "that a|b if and only if for any prime in the factorization of a or b, its exponent in the factorization of a is less than or equal to its exponent in the factorization of b."
I tried to use this theorem where k <= m, k <= n, where m and n are exponents of 2 and 5, respectively, in the factorization of 100! but I was not able to work out the math so far.
Any help will be greatly appreciated!
divisibility
edited Nov 28 '18 at 21:56
amWhy
1
asked Nov 28 '18 at 21:51
UMass1234UMass1234
112
$endgroup$
I was able to get to a theorem saying "that a|b if and only if for any prime in the factorization of a or b, its exponent in the factorization of a is less than or equal to its exponent in the factorization of b."
I tried to use this theorem where k <= m, k <= n, where m and n are exponents of 2 and 5, respectively, in the factorization of 100! but I was not able to work out the math so far.
Any help will be greatly appreciated!
divisibility
divisibility
edited Nov 28 '18 at 21:56
amWhy
1
asked Nov 28 '18 at 21:51
UMass1234UMass1234
112
edited Nov 28 '18 at 21:56
amWhy
1
asked Nov 28 '18 at 21:51
UMass1234UMass1234
112
edited Nov 28 '18 at 21:56
amWhy
1
edited Nov 28 '18 at 21:56
amWhy
1
edited Nov 28 '18 at 21:56
amWhy
1
1
asked Nov 28 '18 at 21:51
UMass1234UMass1234
112
asked Nov 28 '18 at 21:51
UMass1234UMass1234
112
asked Nov 28 '18 at 21:51
UMass1234UMass1234
112
112
1
$begingroup$
Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
$endgroup$
– Ross Millikan
Nov 28 '18 at 21:59
add a comment |
1
$begingroup$
Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
$endgroup$
– Ross Millikan
Nov 28 '18 at 21:59
1
1
$begingroup$
Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
$endgroup$
– Ross Millikan
Nov 28 '18 at 21:59
$begingroup$
Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
$endgroup$
– Ross Millikan
Nov 28 '18 at 21:59
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Hint:
You use Legendre's formula: if $p$ is a prime number and $v_p(n)$ denotes the $p$-adic valuation of the natural number $n$, then
$$v_p(n!)=biggllfloorfrac npbiggrrfloor+biggllfloorfrac n{p^2}biggrrfloor+dotsm $$
answered Nov 28 '18 at 22:08
BernardBernard
120k740113
$endgroup$
$begingroup$
thank you for the hint! Appreciate it.
$endgroup$
– UMass1234
Nov 28 '18 at 22:19
add a comment |
$begingroup$
This looks like a homework problem so I'll not give the complete answer.
Instead I invite you to ask:
How many times does the factor $5$ occur in the first 100 natural numbers? (NB remember that $25$, $50$ and $100$ are each divisible by $5^2$.) What about the factor 2?
What power of 10 can you make out of those?
answered Nov 28 '18 at 22:15
timtfjtimtfj
2,050419
$endgroup$
add a comment |
$begingroup$
10 and 100 divide by 10, as do 2*5, 12*15, 22*25, 32*35, 42*45, 52*55, 62*65, 72*75, 82*85, and 92*95. Multiplying these numbers may also produce more pairs of numbers that divide by 10 when multiplied together, but there won't be any that aren't in this list.
answered Nov 28 '18 at 22:00
SethSeth
42812
$endgroup$
add a comment |
Your Answer
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint:
You use Legendre's formula: if $p$ is a prime number and $v_p(n)$ denotes the $p$-adic valuation of the natural number $n$, then
$$v_p(n!)=biggllfloorfrac npbiggrrfloor+biggllfloorfrac n{p^2}biggrrfloor+dotsm $$
answered Nov 28 '18 at 22:08
BernardBernard
120k740113
$endgroup$
$begingroup$
thank you for the hint! Appreciate it.
$endgroup$
– UMass1234
Nov 28 '18 at 22:19
add a comment |
$begingroup$
Hint:
You use Legendre's formula: if $p$ is a prime number and $v_p(n)$ denotes the $p$-adic valuation of the natural number $n$, then
$$v_p(n!)=biggllfloorfrac npbiggrrfloor+biggllfloorfrac n{p^2}biggrrfloor+dotsm $$
answered Nov 28 '18 at 22:08
BernardBernard
120k740113
$endgroup$
$begingroup$
thank you for the hint! Appreciate it.
$endgroup$
– UMass1234
Nov 28 '18 at 22:19
add a comment |
$begingroup$
Hint:
You use Legendre's formula: if $p$ is a prime number and $v_p(n)$ denotes the $p$-adic valuation of the natural number $n$, then
$$v_p(n!)=biggllfloorfrac npbiggrrfloor+biggllfloorfrac n{p^2}biggrrfloor+dotsm $$
answered Nov 28 '18 at 22:08
BernardBernard
120k740113
$endgroup$
Hint:
You use Legendre's formula: if $p$ is a prime number and $v_p(n)$ denotes the $p$-adic valuation of the natural number $n$, then
$$v_p(n!)=biggllfloorfrac npbiggrrfloor+biggllfloorfrac n{p^2}biggrrfloor+dotsm $$
answered Nov 28 '18 at 22:08
BernardBernard
120k740113
answered Nov 28 '18 at 22:08
BernardBernard
120k740113
answered Nov 28 '18 at 22:08
BernardBernard
120k740113
answered Nov 28 '18 at 22:08
BernardBernard
120k740113
120k740113
$begingroup$
thank you for the hint! Appreciate it.
$endgroup$
– UMass1234
Nov 28 '18 at 22:19
add a comment |
$begingroup$
thank you for the hint! Appreciate it.
$endgroup$
– UMass1234
Nov 28 '18 at 22:19
$begingroup$
thank you for the hint! Appreciate it.
$endgroup$
– UMass1234
Nov 28 '18 at 22:19
$begingroup$
thank you for the hint! Appreciate it.
$endgroup$
– UMass1234
Nov 28 '18 at 22:19
add a comment |
$begingroup$
This looks like a homework problem so I'll not give the complete answer.
Instead I invite you to ask:
How many times does the factor $5$ occur in the first 100 natural numbers? (NB remember that $25$, $50$ and $100$ are each divisible by $5^2$.) What about the factor 2?
What power of 10 can you make out of those?
answered Nov 28 '18 at 22:15
timtfjtimtfj
2,050419
$endgroup$
add a comment |
$begingroup$
This looks like a homework problem so I'll not give the complete answer.
Instead I invite you to ask:
How many times does the factor $5$ occur in the first 100 natural numbers? (NB remember that $25$, $50$ and $100$ are each divisible by $5^2$.) What about the factor 2?
What power of 10 can you make out of those?
answered Nov 28 '18 at 22:15
timtfjtimtfj
2,050419
$endgroup$
add a comment |
$begingroup$
This looks like a homework problem so I'll not give the complete answer.
Instead I invite you to ask:
How many times does the factor $5$ occur in the first 100 natural numbers? (NB remember that $25$, $50$ and $100$ are each divisible by $5^2$.) What about the factor 2?
What power of 10 can you make out of those?
answered Nov 28 '18 at 22:15
timtfjtimtfj
2,050419
$endgroup$
This looks like a homework problem so I'll not give the complete answer.
Instead I invite you to ask:
How many times does the factor $5$ occur in the first 100 natural numbers? (NB remember that $25$, $50$ and $100$ are each divisible by $5^2$.) What about the factor 2?
What power of 10 can you make out of those?
answered Nov 28 '18 at 22:15
timtfjtimtfj
2,050419
answered Nov 28 '18 at 22:15
timtfjtimtfj
2,050419
answered Nov 28 '18 at 22:15
timtfjtimtfj
2,050419
answered Nov 28 '18 at 22:15
timtfjtimtfj
2,050419
2,050419
add a comment |
add a comment |
$begingroup$
10 and 100 divide by 10, as do 2*5, 12*15, 22*25, 32*35, 42*45, 52*55, 62*65, 72*75, 82*85, and 92*95. Multiplying these numbers may also produce more pairs of numbers that divide by 10 when multiplied together, but there won't be any that aren't in this list.
answered Nov 28 '18 at 22:00
SethSeth
42812
$endgroup$
add a comment |
$begingroup$
10 and 100 divide by 10, as do 2*5, 12*15, 22*25, 32*35, 42*45, 52*55, 62*65, 72*75, 82*85, and 92*95. Multiplying these numbers may also produce more pairs of numbers that divide by 10 when multiplied together, but there won't be any that aren't in this list.
answered Nov 28 '18 at 22:00
SethSeth
42812
$endgroup$
add a comment |
$begingroup$
10 and 100 divide by 10, as do 2*5, 12*15, 22*25, 32*35, 42*45, 52*55, 62*65, 72*75, 82*85, and 92*95. Multiplying these numbers may also produce more pairs of numbers that divide by 10 when multiplied together, but there won't be any that aren't in this list.
answered Nov 28 '18 at 22:00
SethSeth
42812
$endgroup$
10 and 100 divide by 10, as do 2*5, 12*15, 22*25, 32*35, 42*45, 52*55, 62*65, 72*75, 82*85, and 92*95. Multiplying these numbers may also produce more pairs of numbers that divide by 10 when multiplied together, but there won't be any that aren't in this list.
answered Nov 28 '18 at 22:00
SethSeth
42812
answered Nov 28 '18 at 22:00
SethSeth
42812
answered Nov 28 '18 at 22:00
SethSeth
42812
answered Nov 28 '18 at 22:00
SethSeth
42812
42812
add a comment |
add a comment |
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$begingroup$
Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
$endgroup$
– Ross Millikan
Nov 28 '18 at 21:59