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







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  • 1




    Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
    – Ross Millikan
    Nov 28 at 21:59















up vote
-1
down vote

favorite












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







share|cite|improve this question




















  • 1




    Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
    – Ross Millikan
    Nov 28 at 21:59













up vote
-1
down vote

favorite









up vote
-1
down vote

favorite











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







share|cite|improve this question















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






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 28 at 21:56









amWhy

191k28224439




191k28224439










asked Nov 28 at 21:51









UMass1234

112




112








  • 1




    Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
    – Ross Millikan
    Nov 28 at 21:59














  • 1




    Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
    – Ross Millikan
    Nov 28 at 21:59








1




1




Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
– Ross Millikan
Nov 28 at 21:59




Do you know Legendre's formula for the power of a prime dividing a factorial? You can also search the site for this.
– Ross Millikan
Nov 28 at 21:59










3 Answers
3






active

oldest

votes

















up vote
0
down vote



accepted










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



$$v_p(n!)=biggllfloorfrac npbiggrrfloor+biggllfloorfrac n{p^2}biggrrfloor+dotsm $$






share|cite|improve this answer





















  • thank you for the hint! Appreciate it.
    – UMass1234
    Nov 28 at 22:19


















up vote
1
down vote













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?






share|cite|improve this answer




























    up vote
    0
    down vote













    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.






    share|cite|improve this answer





















      Your Answer





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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      0
      down vote



      accepted










      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



      $$v_p(n!)=biggllfloorfrac npbiggrrfloor+biggllfloorfrac n{p^2}biggrrfloor+dotsm $$






      share|cite|improve this answer





















      • thank you for the hint! Appreciate it.
        – UMass1234
        Nov 28 at 22:19















      up vote
      0
      down vote



      accepted










      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



      $$v_p(n!)=biggllfloorfrac npbiggrrfloor+biggllfloorfrac n{p^2}biggrrfloor+dotsm $$






      share|cite|improve this answer





















      • thank you for the hint! Appreciate it.
        – UMass1234
        Nov 28 at 22:19













      up vote
      0
      down vote



      accepted







      up vote
      0
      down vote



      accepted






      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



      $$v_p(n!)=biggllfloorfrac npbiggrrfloor+biggllfloorfrac n{p^2}biggrrfloor+dotsm $$






      share|cite|improve this answer












      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



      $$v_p(n!)=biggllfloorfrac npbiggrrfloor+biggllfloorfrac n{p^2}biggrrfloor+dotsm $$







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered Nov 28 at 22:08









      Bernard

      117k637110




      117k637110












      • thank you for the hint! Appreciate it.
        – UMass1234
        Nov 28 at 22:19


















      • thank you for the hint! Appreciate it.
        – UMass1234
        Nov 28 at 22:19
















      thank you for the hint! Appreciate it.
      – UMass1234
      Nov 28 at 22:19




      thank you for the hint! Appreciate it.
      – UMass1234
      Nov 28 at 22:19










      up vote
      1
      down vote













      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?






      share|cite|improve this answer

























        up vote
        1
        down vote













        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?






        share|cite|improve this answer























          up vote
          1
          down vote










          up vote
          1
          down vote









          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?






          share|cite|improve this answer












          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?







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 28 at 22:15









          timtfj

          879217




          879217






















              up vote
              0
              down vote













              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.






              share|cite|improve this answer

























                up vote
                0
                down vote













                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.






                share|cite|improve this answer























                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  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.






                  share|cite|improve this answer












                  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.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 28 at 22:00









                  Seth

                  43612




                  43612






























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