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.
contest-math
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." â user21820, TheSimpliFire, user91500, HK Lee, amWhy
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up vote
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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.
contest-math
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." â user21820, TheSimpliFire, user91500, HK Lee, amWhy
@TheSimpliFire Thanks, I just edited the question to be more specific.
â Vivek Sivaramakrishnan
Sep 2 at 8:48
related/duplicate: 2013, 2015, 2016, 2018
â farruhota
Sep 2 at 9:59
Since you know that $a=b$ why don't you just write it as $aaaa$?
â bof
Sep 2 at 10:02
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I want to solve the above question systematically, i.e, assuming that I do not know all the $4$-digit square numbers.
contest-math
I want to solve the above question systematically, i.e, assuming that I do not know all the $4$-digit square numbers.
contest-math
contest-math
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." â user21820, TheSimpliFire, user91500, HK Lee, amWhy
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." â user21820, TheSimpliFire, user91500, HK Lee, amWhy
@TheSimpliFire Thanks, I just edited the question to be more specific.
â Vivek Sivaramakrishnan
Sep 2 at 8:48
related/duplicate: 2013, 2015, 2016, 2018
â farruhota
Sep 2 at 9:59
Since you know that $a=b$ why don't you just write it as $aaaa$?
â bof
Sep 2 at 10:02
add a comment |Â
@TheSimpliFire Thanks, I just edited the question to be more specific.
â Vivek Sivaramakrishnan
Sep 2 at 8:48
related/duplicate: 2013, 2015, 2016, 2018
â farruhota
Sep 2 at 9:59
Since you know that $a=b$ why don't you just write it as $aaaa$?
â bof
Sep 2 at 10:02
@TheSimpliFire Thanks, I just edited the question to be more specific.
â Vivek Sivaramakrishnan
Sep 2 at 8:48
@TheSimpliFire Thanks, I just edited the question to be more specific.
â Vivek Sivaramakrishnan
Sep 2 at 8:48
related/duplicate: 2013, 2015, 2016, 2018
â farruhota
Sep 2 at 9:59
related/duplicate: 2013, 2015, 2016, 2018
â farruhota
Sep 2 at 9:59
Since you know that $a=b$ why don't you just write it as $aaaa$?
â bof
Sep 2 at 10:02
Since you know that $a=b$ why don't you just write it as $aaaa$?
â bof
Sep 2 at 10:02
add a comment |Â
2 Answers
2
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oldest
votes
up vote
1
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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
- $0000,7744$
squares don't end in $3$, $7$ or $8$.
â Lord Shark the Unknown
Sep 2 at 9:37
@LordSharktheUnknown Yes of course! I fix that. Thanks
â gimusi
Sep 2 at 9:45
add a comment |Â
up vote
1
down vote
Hint: Remember that your number can be written in the form
$$z=b+10b+100a+1000a=11b+1100a$$
add a comment |Â
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
- $0000,7744$
squares don't end in $3$, $7$ or $8$.
â Lord Shark the Unknown
Sep 2 at 9:37
@LordSharktheUnknown Yes of course! I fix that. Thanks
â gimusi
Sep 2 at 9:45
add a comment |Â
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
- $0000,7744$
squares don't end in $3$, $7$ or $8$.
â Lord Shark the Unknown
Sep 2 at 9:37
@LordSharktheUnknown Yes of course! I fix that. Thanks
â gimusi
Sep 2 at 9:45
add a comment |Â
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
- $0000,7744$
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
- $0000,7744$
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 $8$.
â Lord Shark the Unknown
Sep 2 at 9:37
@LordSharktheUnknown Yes of course! I fix that. Thanks
â gimusi
Sep 2 at 9:45
add a comment |Â
squares don't end in $3$, $7$ or $8$.
â Lord Shark the Unknown
Sep 2 at 9:37
@LordSharktheUnknown Yes of course! I fix that. Thanks
â gimusi
Sep 2 at 9:45
squares don't end in $3$, $7$ or $8$.
â Lord Shark the Unknown
Sep 2 at 9:37
squares don't end in $3$, $7$ or $8$.
â Lord Shark the Unknown
Sep 2 at 9:37
@LordSharktheUnknown Yes of course! I fix that. Thanks
â gimusi
Sep 2 at 9:45
@LordSharktheUnknown Yes of course! I fix that. Thanks
â gimusi
Sep 2 at 9:45
add a comment |Â
up vote
1
down vote
Hint: Remember that your number can be written in the form
$$z=b+10b+100a+1000a=11b+1100a$$
add a comment |Â
up vote
1
down vote
Hint: Remember that your number can be written in the form
$$z=b+10b+100a+1000a=11b+1100a$$
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Hint: Remember that your number can be written in the form
$$z=b+10b+100a+1000a=11b+1100a$$
Hint: Remember that your number can be written in the form
$$z=b+10b+100a+1000a=11b+1100a$$
answered Sep 2 at 8:46
Dr. Sonnhard Graubner
68.6k32760
68.6k32760
add a comment |Â
add a comment |Â
@TheSimpliFire Thanks, I just edited the question to be more specific.
â Vivek Sivaramakrishnan
Sep 2 at 8:48
related/duplicate: 2013, 2015, 2016, 2018
â farruhota
Sep 2 at 9:59
Since you know that $a=b$ why don't you just write it as $aaaa$?
â bof
Sep 2 at 10:02