Given $aabb$ is a square number, and $a := b$, find $a$ and $b$. [closed]

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I want to solve the above question systematically, i.e, assuming that I do not know all the $4$-digit square numbers.










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closed as off-topic by user21820, TheSimpliFire, user91500, HK Lee, amWhy Sep 2 at 10:52


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." – user21820, TheSimpliFire, user91500, HK Lee, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.












  • @TheSimpliFire Thanks, I just edited the question to be more specific.
    – Vivek Sivaramakrishnan
    Sep 2 at 8:48











  • related/duplicate: 2013, 2015, 2016, 2018
    – farruhota
    Sep 2 at 9:59










  • Since you know that $a=b$ why don't you just write it as $aaaa$?
    – bof
    Sep 2 at 10:02














up vote
0
down vote

favorite
1












I want to solve the above question systematically, i.e, assuming that I do not know all the $4$-digit square numbers.










share|cite|improve this question















closed as off-topic by user21820, TheSimpliFire, user91500, HK Lee, amWhy Sep 2 at 10:52


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." – user21820, TheSimpliFire, user91500, HK Lee, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.












  • @TheSimpliFire Thanks, I just edited the question to be more specific.
    – Vivek Sivaramakrishnan
    Sep 2 at 8:48











  • related/duplicate: 2013, 2015, 2016, 2018
    – farruhota
    Sep 2 at 9:59










  • Since you know that $a=b$ why don't you just write it as $aaaa$?
    – bof
    Sep 2 at 10:02












up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





I want to solve the above question systematically, i.e, assuming that I do not know all the $4$-digit square numbers.










share|cite|improve this question















I want to solve the above question systematically, i.e, assuming that I do not know all the $4$-digit square numbers.







contest-math






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edited Sep 2 at 9:40









José Carlos Santos

121k16101186




121k16101186










asked Sep 2 at 8:40









Vivek Sivaramakrishnan

314




314




closed as off-topic by user21820, TheSimpliFire, user91500, HK Lee, amWhy Sep 2 at 10:52


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." – user21820, TheSimpliFire, user91500, HK Lee, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by user21820, TheSimpliFire, user91500, HK Lee, amWhy Sep 2 at 10:52


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." – user21820, TheSimpliFire, user91500, HK Lee, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.











  • @TheSimpliFire Thanks, I just edited the question to be more specific.
    – Vivek Sivaramakrishnan
    Sep 2 at 8:48











  • related/duplicate: 2013, 2015, 2016, 2018
    – farruhota
    Sep 2 at 9:59










  • Since you know that $a=b$ why don't you just write it as $aaaa$?
    – bof
    Sep 2 at 10:02
















  • @TheSimpliFire Thanks, I just edited the question to be more specific.
    – Vivek Sivaramakrishnan
    Sep 2 at 8:48











  • related/duplicate: 2013, 2015, 2016, 2018
    – farruhota
    Sep 2 at 9:59










  • Since you know that $a=b$ why don't you just write it as $aaaa$?
    – bof
    Sep 2 at 10:02















@TheSimpliFire Thanks, I just edited the question to be more specific.
– Vivek Sivaramakrishnan
Sep 2 at 8:48





@TheSimpliFire Thanks, I just edited the question to be more specific.
– Vivek Sivaramakrishnan
Sep 2 at 8:48













related/duplicate: 2013, 2015, 2016, 2018
– farruhota
Sep 2 at 9:59




related/duplicate: 2013, 2015, 2016, 2018
– farruhota
Sep 2 at 9:59












Since you know that $a=b$ why don't you just write it as $aaaa$?
– bof
Sep 2 at 10:02




Since you know that $a=b$ why don't you just write it as $aaaa$?
– bof
Sep 2 at 10:02










2 Answers
2






active

oldest

votes

















up vote
1
down vote



accepted










We have that



$$aabb=1100a+11b=11(100a+b)$$



then we need



$$11|100a+b iff a+bequiv0 pmod11$$



moreover



$$1100a+11bequiv 0,1 pmod 4 iff 3b equiv 0,1 pmod 4 iff b equiv 0,3 pmod 4$$



but since squares doesn't end with $3$, $7$ or $8$ then we need to check among



  • $0000,7744$





share|cite|improve this answer






















  • squares don't end in $3$, $7$ or $8$.
    – Lord Shark the Unknown
    Sep 2 at 9:37










  • @LordSharktheUnknown Yes of course! I fix that. Thanks
    – gimusi
    Sep 2 at 9:45

















up vote
1
down vote













Hint: Remember that your number can be written in the form
$$z=b+10b+100a+1000a=11b+1100a$$






share|cite|improve this answer



























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote



    accepted










    We have that



    $$aabb=1100a+11b=11(100a+b)$$



    then we need



    $$11|100a+b iff a+bequiv0 pmod11$$



    moreover



    $$1100a+11bequiv 0,1 pmod 4 iff 3b equiv 0,1 pmod 4 iff b equiv 0,3 pmod 4$$



    but since squares doesn't end with $3$, $7$ or $8$ then we need to check among



    • $0000,7744$





    share|cite|improve this answer






















    • squares don't end in $3$, $7$ or $8$.
      – Lord Shark the Unknown
      Sep 2 at 9:37










    • @LordSharktheUnknown Yes of course! I fix that. Thanks
      – gimusi
      Sep 2 at 9:45














    up vote
    1
    down vote



    accepted










    We have that



    $$aabb=1100a+11b=11(100a+b)$$



    then we need



    $$11|100a+b iff a+bequiv0 pmod11$$



    moreover



    $$1100a+11bequiv 0,1 pmod 4 iff 3b equiv 0,1 pmod 4 iff b equiv 0,3 pmod 4$$



    but since squares doesn't end with $3$, $7$ or $8$ then we need to check among



    • $0000,7744$





    share|cite|improve this answer






















    • squares don't end in $3$, $7$ or $8$.
      – Lord Shark the Unknown
      Sep 2 at 9:37










    • @LordSharktheUnknown Yes of course! I fix that. Thanks
      – gimusi
      Sep 2 at 9:45












    up vote
    1
    down vote



    accepted







    up vote
    1
    down vote



    accepted






    We have that



    $$aabb=1100a+11b=11(100a+b)$$



    then we need



    $$11|100a+b iff a+bequiv0 pmod11$$



    moreover



    $$1100a+11bequiv 0,1 pmod 4 iff 3b equiv 0,1 pmod 4 iff b equiv 0,3 pmod 4$$



    but since squares doesn't end with $3$, $7$ or $8$ then we need to check among



    • $0000,7744$





    share|cite|improve this answer














    We have that



    $$aabb=1100a+11b=11(100a+b)$$



    then we need



    $$11|100a+b iff a+bequiv0 pmod11$$



    moreover



    $$1100a+11bequiv 0,1 pmod 4 iff 3b equiv 0,1 pmod 4 iff b equiv 0,3 pmod 4$$



    but since squares doesn't end with $3$, $7$ or $8$ then we need to check among



    • $0000,7744$






    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Sep 2 at 9:48

























    answered Sep 2 at 8:44









    gimusi

    72.2k73888




    72.2k73888











    • squares don't end in $3$, $7$ or $8$.
      – Lord Shark the Unknown
      Sep 2 at 9:37










    • @LordSharktheUnknown Yes of course! I fix that. Thanks
      – gimusi
      Sep 2 at 9:45
















    • squares don't end in $3$, $7$ or $8$.
      – Lord Shark the Unknown
      Sep 2 at 9:37










    • @LordSharktheUnknown Yes of course! I fix that. Thanks
      – gimusi
      Sep 2 at 9:45















    squares don't end in $3$, $7$ or $8$.
    – Lord Shark the Unknown
    Sep 2 at 9:37




    squares don't end in $3$, $7$ or $8$.
    – Lord Shark the Unknown
    Sep 2 at 9:37












    @LordSharktheUnknown Yes of course! I fix that. Thanks
    – gimusi
    Sep 2 at 9:45




    @LordSharktheUnknown Yes of course! I fix that. Thanks
    – gimusi
    Sep 2 at 9:45










    up vote
    1
    down vote













    Hint: Remember that your number can be written in the form
    $$z=b+10b+100a+1000a=11b+1100a$$






    share|cite|improve this answer
























      up vote
      1
      down vote













      Hint: Remember that your number can be written in the form
      $$z=b+10b+100a+1000a=11b+1100a$$






      share|cite|improve this answer






















        up vote
        1
        down vote










        up vote
        1
        down vote









        Hint: Remember that your number can be written in the form
        $$z=b+10b+100a+1000a=11b+1100a$$






        share|cite|improve this answer












        Hint: Remember that your number can be written in the form
        $$z=b+10b+100a+1000a=11b+1100a$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Sep 2 at 8:46









        Dr. Sonnhard Graubner

        68.6k32760




        68.6k32760












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