Smallest positive integer that is divisble by 2, 3 and 5 and that is also a perfect square and perfect cube...












0














My son is having this problem:




Smallest positive integer that is divisible by $2$, $3$ and $5$, and that is also a perfect square and perfect cube.




He was busy using Excel, but I thought: Isn't there a faster way?






number-theory elementary-number-theory







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closed as off-topic by José Carlos Santos, GEdgar, Adrian Keister, amWhy, Rebellos Nov 20 '18 at 17:49


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, GEdgar, Adrian Keister, amWhy, Rebellos

If this question can be reworded to fit the rules in the help center, please edit the question.













  • $$(2cdot3cdot5)^{6m}$$
    – lab bhattacharjee
    Nov 20 '18 at 12:13










  • @labbhattacharjee: Since he said "smallest", my answer would be $(2cdot 3cdot 5)^6$,
    – GEdgar
    Nov 20 '18 at 12:16
















0














My son is having this problem:




Smallest positive integer that is divisible by $2$, $3$ and $5$, and that is also a perfect square and perfect cube.




He was busy using Excel, but I thought: Isn't there a faster way?






number-theory elementary-number-theory







share|cite|improve this question















closed as off-topic by José Carlos Santos, GEdgar, Adrian Keister, amWhy, Rebellos Nov 20 '18 at 17:49


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, GEdgar, Adrian Keister, amWhy, Rebellos

If this question can be reworded to fit the rules in the help center, please edit the question.













  • $$(2cdot3cdot5)^{6m}$$
    – lab bhattacharjee
    Nov 20 '18 at 12:13










  • @labbhattacharjee: Since he said "smallest", my answer would be $(2cdot 3cdot 5)^6$,
    – GEdgar
    Nov 20 '18 at 12:16














0












0








0







My son is having this problem:




Smallest positive integer that is divisible by $2$, $3$ and $5$, and that is also a perfect square and perfect cube.




He was busy using Excel, but I thought: Isn't there a faster way?






number-theory elementary-number-theory







share|cite|improve this question















My son is having this problem:




Smallest positive integer that is divisible by $2$, $3$ and $5$, and that is also a perfect square and perfect cube.




He was busy using Excel, but I thought: Isn't there a faster way?






number-theory elementary-number-theory




number-theory elementary-number-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 20 '18 at 12:31









cansomeonehelpmeout

6,7483835




6,7483835










asked Nov 20 '18 at 12:12









mathpieuler

13610




13610




closed as off-topic by José Carlos Santos, GEdgar, Adrian Keister, amWhy, Rebellos Nov 20 '18 at 17:49


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, GEdgar, Adrian Keister, amWhy, Rebellos

If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by José Carlos Santos, GEdgar, Adrian Keister, amWhy, Rebellos Nov 20 '18 at 17:49


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, GEdgar, Adrian Keister, amWhy, Rebellos

If this question can be reworded to fit the rules in the help center, please edit the question.












  • $$(2cdot3cdot5)^{6m}$$
    – lab bhattacharjee
    Nov 20 '18 at 12:13










  • @labbhattacharjee: Since he said "smallest", my answer would be $(2cdot 3cdot 5)^6$,
    – GEdgar
    Nov 20 '18 at 12:16


















  • $$(2cdot3cdot5)^{6m}$$
    – lab bhattacharjee
    Nov 20 '18 at 12:13










  • @labbhattacharjee: Since he said "smallest", my answer would be $(2cdot 3cdot 5)^6$,
    – GEdgar
    Nov 20 '18 at 12:16
















$$(2cdot3cdot5)^{6m}$$
– lab bhattacharjee
Nov 20 '18 at 12:13




$$(2cdot3cdot5)^{6m}$$
– lab bhattacharjee
Nov 20 '18 at 12:13












@labbhattacharjee: Since he said "smallest", my answer would be $(2cdot 3cdot 5)^6$,
– GEdgar
Nov 20 '18 at 12:16




@labbhattacharjee: Since he said "smallest", my answer would be $(2cdot 3cdot 5)^6$,
– GEdgar
Nov 20 '18 at 12:16










2 Answers
2






active

oldest

votes


















1














If $n$ is an integer, then





  • $n^2$ is a perfect square


  • $n^3$ is a perfect cube


An integer that has both these properties is $n^{2cdot 3}=n^{6}$, since $$n^{6}=underset{color{red}{text{square}}}{underbrace{(n^{3})^color{red}{2}}}=underset{color{green}{text{cube}}}{underbrace{(n^{2})^color{green}{3}}}$$ You also want this to be divisible by $2,3,5$, these are prime numbers so the smallest non-zero such number is $$(2cdot 3cdot 5)^6=30^6=729 000 000$$






share|cite|improve this answer





























    2














    Since number has to be divisible by $2,3,5$ therefore it should be divisible by $30$.
    Since you want a perfect square and a perfect cube therefore your number should be of the form $30^{6t}$ . Where t is the integral number satisfying all such conditions.
    In your case $t=1$ , therefore answer is $30^6$.






    share|cite|improve this answer





















    • It took me a long time to edit 😥, I could have posted it before lab bhattacharjee.
      – Akash Roy
      Nov 20 '18 at 12:21










    • Thanks for upvote , next time I will try my best not to delay.
      – Akash Roy
      Nov 20 '18 at 12:38


















    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1














    If $n$ is an integer, then





    • $n^2$ is a perfect square


    • $n^3$ is a perfect cube


    An integer that has both these properties is $n^{2cdot 3}=n^{6}$, since $$n^{6}=underset{color{red}{text{square}}}{underbrace{(n^{3})^color{red}{2}}}=underset{color{green}{text{cube}}}{underbrace{(n^{2})^color{green}{3}}}$$ You also want this to be divisible by $2,3,5$, these are prime numbers so the smallest non-zero such number is $$(2cdot 3cdot 5)^6=30^6=729 000 000$$






    share|cite|improve this answer


























      1














      If $n$ is an integer, then





      • $n^2$ is a perfect square


      • $n^3$ is a perfect cube


      An integer that has both these properties is $n^{2cdot 3}=n^{6}$, since $$n^{6}=underset{color{red}{text{square}}}{underbrace{(n^{3})^color{red}{2}}}=underset{color{green}{text{cube}}}{underbrace{(n^{2})^color{green}{3}}}$$ You also want this to be divisible by $2,3,5$, these are prime numbers so the smallest non-zero such number is $$(2cdot 3cdot 5)^6=30^6=729 000 000$$






      share|cite|improve this answer
























        1












        1








        1






        If $n$ is an integer, then





        • $n^2$ is a perfect square


        • $n^3$ is a perfect cube


        An integer that has both these properties is $n^{2cdot 3}=n^{6}$, since $$n^{6}=underset{color{red}{text{square}}}{underbrace{(n^{3})^color{red}{2}}}=underset{color{green}{text{cube}}}{underbrace{(n^{2})^color{green}{3}}}$$ You also want this to be divisible by $2,3,5$, these are prime numbers so the smallest non-zero such number is $$(2cdot 3cdot 5)^6=30^6=729 000 000$$






        share|cite|improve this answer












        If $n$ is an integer, then





        • $n^2$ is a perfect square


        • $n^3$ is a perfect cube


        An integer that has both these properties is $n^{2cdot 3}=n^{6}$, since $$n^{6}=underset{color{red}{text{square}}}{underbrace{(n^{3})^color{red}{2}}}=underset{color{green}{text{cube}}}{underbrace{(n^{2})^color{green}{3}}}$$ You also want this to be divisible by $2,3,5$, these are prime numbers so the smallest non-zero such number is $$(2cdot 3cdot 5)^6=30^6=729 000 000$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 20 '18 at 12:23









        cansomeonehelpmeout

        6,7483835




        6,7483835























            2














            Since number has to be divisible by $2,3,5$ therefore it should be divisible by $30$.
            Since you want a perfect square and a perfect cube therefore your number should be of the form $30^{6t}$ . Where t is the integral number satisfying all such conditions.
            In your case $t=1$ , therefore answer is $30^6$.






            share|cite|improve this answer





















            • It took me a long time to edit 😥, I could have posted it before lab bhattacharjee.
              – Akash Roy
              Nov 20 '18 at 12:21










            • Thanks for upvote , next time I will try my best not to delay.
              – Akash Roy
              Nov 20 '18 at 12:38
















            2














            Since number has to be divisible by $2,3,5$ therefore it should be divisible by $30$.
            Since you want a perfect square and a perfect cube therefore your number should be of the form $30^{6t}$ . Where t is the integral number satisfying all such conditions.
            In your case $t=1$ , therefore answer is $30^6$.






            share|cite|improve this answer





















            • It took me a long time to edit 😥, I could have posted it before lab bhattacharjee.
              – Akash Roy
              Nov 20 '18 at 12:21










            • Thanks for upvote , next time I will try my best not to delay.
              – Akash Roy
              Nov 20 '18 at 12:38














            2












            2








            2






            Since number has to be divisible by $2,3,5$ therefore it should be divisible by $30$.
            Since you want a perfect square and a perfect cube therefore your number should be of the form $30^{6t}$ . Where t is the integral number satisfying all such conditions.
            In your case $t=1$ , therefore answer is $30^6$.






            share|cite|improve this answer












            Since number has to be divisible by $2,3,5$ therefore it should be divisible by $30$.
            Since you want a perfect square and a perfect cube therefore your number should be of the form $30^{6t}$ . Where t is the integral number satisfying all such conditions.
            In your case $t=1$ , therefore answer is $30^6$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Nov 20 '18 at 12:20









            Akash Roy

            1




            1












            • It took me a long time to edit 😥, I could have posted it before lab bhattacharjee.
              – Akash Roy
              Nov 20 '18 at 12:21










            • Thanks for upvote , next time I will try my best not to delay.
              – Akash Roy
              Nov 20 '18 at 12:38


















            • It took me a long time to edit 😥, I could have posted it before lab bhattacharjee.
              – Akash Roy
              Nov 20 '18 at 12:21










            • Thanks for upvote , next time I will try my best not to delay.
              – Akash Roy
              Nov 20 '18 at 12:38
















            It took me a long time to edit 😥, I could have posted it before lab bhattacharjee.
            – Akash Roy
            Nov 20 '18 at 12:21




            It took me a long time to edit 😥, I could have posted it before lab bhattacharjee.
            – Akash Roy
            Nov 20 '18 at 12:21












            Thanks for upvote , next time I will try my best not to delay.
            – Akash Roy
            Nov 20 '18 at 12:38




            Thanks for upvote , next time I will try my best not to delay.
            – Akash Roy
            Nov 20 '18 at 12:38



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