Mixing
![How can I get a programming job abroad with some experience but no formal computing qualifications? [closed]](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjbpfN9tAutmK93bJRC3ZoROZzi2TJDms5n8_qJuhgE0a9b52OOHayv3NGT8igAdFL7byXNst-_1DZK5SjrIJ28_6RQPUpBROqMs5s6jo-ZsjX8kjDwfxJufIitH3TaQRXWaGSQKRQib-f/s72-c/1.jpg)
What is the smallest integer greater than 1 such that 1/2 of it is a perfect square and 1/5 of it is a perfect fifth power?
Clash Royale CLAN TAG#URR8PPP
up vote
4
down vote
favorite
What is the smallest integer greater than 1 such that 1/2 of it is a perfect square and 1/5 of it is a perfect fifth power?
I have tried multiplying every perfect square (up to 400 by two and checking if it is a perfect 5th power, but still nothing. I don't know what to do at this point.
integers
asked 28 mins ago
J. DOEE
211
New contributor
J. DOEE is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
4
down vote
favorite
What is the smallest integer greater than 1 such that 1/2 of it is a perfect square and 1/5 of it is a perfect fifth power?
I have tried multiplying every perfect square (up to 400 by two and checking if it is a perfect 5th power, but still nothing. I don't know what to do at this point.
integers
asked 28 mins ago
J. DOEE
211
New contributor
J. DOEE is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
3
It has to be divisible by $5$ and $2$ so you are looking for a multiple of $10$. Working with perfect fifths will get you through the numbers faster. The number $500000$ works so you can work down from there.
– John Douma
23 mins ago
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
What is the smallest integer greater than 1 such that 1/2 of it is a perfect square and 1/5 of it is a perfect fifth power?
I have tried multiplying every perfect square (up to 400 by two and checking if it is a perfect 5th power, but still nothing. I don't know what to do at this point.
integers
asked 28 mins ago
J. DOEE
211
New contributor
J. DOEE is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
What is the smallest integer greater than 1 such that 1/2 of it is a perfect square and 1/5 of it is a perfect fifth power?
I have tried multiplying every perfect square (up to 400 by two and checking if it is a perfect 5th power, but still nothing. I don't know what to do at this point.
integers
integers
asked 28 mins ago
J. DOEE
211
New contributor
J. DOEE is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 28 mins ago
J. DOEE
211
New contributor
J. DOEE is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 28 mins ago
J. DOEE
211
New contributor
J. DOEE is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 28 mins ago
J. DOEE
211
asked 28 mins ago
J. DOEE
211
211
New contributor
J. DOEE is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
J. DOEE is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
J. DOEE is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
3
It has to be divisible by $5$ and $2$ so you are looking for a multiple of $10$. Working with perfect fifths will get you through the numbers faster. The number $500000$ works so you can work down from there.
– John Douma
23 mins ago
add a comment |Â
3
It has to be divisible by $5$ and $2$ so you are looking for a multiple of $10$. Working with perfect fifths will get you through the numbers faster. The number $500000$ works so you can work down from there.
– John Douma
23 mins ago
3
3
It has to be divisible by $5$ and $2$ so you are looking for a multiple of $10$. Working with perfect fifths will get you through the numbers faster. The number $500000$ works so you can work down from there.
– John Douma
23 mins ago
It has to be divisible by $5$ and $2$ so you are looking for a multiple of $10$. Working with perfect fifths will get you through the numbers faster. The number $500000$ works so you can work down from there.
– John Douma
23 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
6
down vote
The number is clearly a multiple of $5$ and $2$. We look for the smallest, so we assume that it has no more prime factors.
So let $n=2^a5^b$. Since $n/2$ is a square, then $a-1$ and $b$ are even. Since $n/5$ is a fifth power, $a$ and $b-1$ are multiples of $5$. Then $a=5$ and $b=6$.
answered 19 mins ago
ajotatxe
51k13185
1
Why can you assume that it has no other prime factors?
– Servaes
12 mins ago
add a comment |Â
up vote
1
down vote
Here's a very unsophisticated approach: Let $n$ be the smallest such integer. Then there exist integers $a$ and $b$ such that $n=5a^5$ and $n=2b^2$. It follows that $a$ is a multiple of $2$, say $a=2a_1$, and $b$ is a multiple of $5$, say $b=5b_1$. Then
$$n=2^5cdot5cdot a_1^5qquadtext and qquad n=2cdot5^2cdot b_1^2.$$
This in turn shows that $a_1$ is a multiple of $5$, say $a_1=5a_2$, and $b_1$ is a multiple of $2$, say $b_1=2b_2$. Then
$$n=2^5cdot5^6cdot a_2^5qquadtext and qquad n=2^3cdot5^2cdot b_2^2.$$
This in turn shows that $b_2$ is a multiple of both $2$ and $5^2$, say $b_2=2cdot5^2cdot b_3$. Then
$$n=2^5cdot5^6cdot a_2^5qquadtext and qquad n=2^5cdot5^6cdot b_3^2.$$
This shows that $ngeq2^5cdot5^6$, and as you might expect a quick check shows that $n=2^5cdot5^6$ does indeed work, so $n=2^5cdot5^6=500000$.
answered 12 mins ago
Servaes
19.5k33686
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
The number is clearly a multiple of $5$ and $2$. We look for the smallest, so we assume that it has no more prime factors.
So let $n=2^a5^b$. Since $n/2$ is a square, then $a-1$ and $b$ are even. Since $n/5$ is a fifth power, $a$ and $b-1$ are multiples of $5$. Then $a=5$ and $b=6$.
answered 19 mins ago
ajotatxe
51k13185
1
Why can you assume that it has no other prime factors?
– Servaes
12 mins ago
add a comment |Â
up vote
6
down vote
The number is clearly a multiple of $5$ and $2$. We look for the smallest, so we assume that it has no more prime factors.
So let $n=2^a5^b$. Since $n/2$ is a square, then $a-1$ and $b$ are even. Since $n/5$ is a fifth power, $a$ and $b-1$ are multiples of $5$. Then $a=5$ and $b=6$.
answered 19 mins ago
ajotatxe
51k13185
1
Why can you assume that it has no other prime factors?
– Servaes
12 mins ago
add a comment |Â
up vote
6
down vote
up vote
6
down vote
The number is clearly a multiple of $5$ and $2$. We look for the smallest, so we assume that it has no more prime factors.
So let $n=2^a5^b$. Since $n/2$ is a square, then $a-1$ and $b$ are even. Since $n/5$ is a fifth power, $a$ and $b-1$ are multiples of $5$. Then $a=5$ and $b=6$.
answered 19 mins ago
ajotatxe
51k13185
The number is clearly a multiple of $5$ and $2$. We look for the smallest, so we assume that it has no more prime factors.
So let $n=2^a5^b$. Since $n/2$ is a square, then $a-1$ and $b$ are even. Since $n/5$ is a fifth power, $a$ and $b-1$ are multiples of $5$. Then $a=5$ and $b=6$.
answered 19 mins ago
ajotatxe
51k13185
answered 19 mins ago
ajotatxe
51k13185
answered 19 mins ago
ajotatxe
51k13185
answered 19 mins ago
ajotatxe
51k13185
51k13185
1
Why can you assume that it has no other prime factors?
– Servaes
12 mins ago
add a comment |Â
1
Why can you assume that it has no other prime factors?
– Servaes
12 mins ago
1
1
Why can you assume that it has no other prime factors?
– Servaes
12 mins ago
Why can you assume that it has no other prime factors?
– Servaes
12 mins ago
add a comment |Â
up vote
1
down vote
Here's a very unsophisticated approach: Let $n$ be the smallest such integer. Then there exist integers $a$ and $b$ such that $n=5a^5$ and $n=2b^2$. It follows that $a$ is a multiple of $2$, say $a=2a_1$, and $b$ is a multiple of $5$, say $b=5b_1$. Then
$$n=2^5cdot5cdot a_1^5qquadtext and qquad n=2cdot5^2cdot b_1^2.$$
This in turn shows that $a_1$ is a multiple of $5$, say $a_1=5a_2$, and $b_1$ is a multiple of $2$, say $b_1=2b_2$. Then
$$n=2^5cdot5^6cdot a_2^5qquadtext and qquad n=2^3cdot5^2cdot b_2^2.$$
This in turn shows that $b_2$ is a multiple of both $2$ and $5^2$, say $b_2=2cdot5^2cdot b_3$. Then
$$n=2^5cdot5^6cdot a_2^5qquadtext and qquad n=2^5cdot5^6cdot b_3^2.$$
This shows that $ngeq2^5cdot5^6$, and as you might expect a quick check shows that $n=2^5cdot5^6$ does indeed work, so $n=2^5cdot5^6=500000$.
answered 12 mins ago
Servaes
19.5k33686
add a comment |Â
up vote
1
down vote
Here's a very unsophisticated approach: Let $n$ be the smallest such integer. Then there exist integers $a$ and $b$ such that $n=5a^5$ and $n=2b^2$. It follows that $a$ is a multiple of $2$, say $a=2a_1$, and $b$ is a multiple of $5$, say $b=5b_1$. Then
$$n=2^5cdot5cdot a_1^5qquadtext and qquad n=2cdot5^2cdot b_1^2.$$
This in turn shows that $a_1$ is a multiple of $5$, say $a_1=5a_2$, and $b_1$ is a multiple of $2$, say $b_1=2b_2$. Then
$$n=2^5cdot5^6cdot a_2^5qquadtext and qquad n=2^3cdot5^2cdot b_2^2.$$
This in turn shows that $b_2$ is a multiple of both $2$ and $5^2$, say $b_2=2cdot5^2cdot b_3$. Then
$$n=2^5cdot5^6cdot a_2^5qquadtext and qquad n=2^5cdot5^6cdot b_3^2.$$
This shows that $ngeq2^5cdot5^6$, and as you might expect a quick check shows that $n=2^5cdot5^6$ does indeed work, so $n=2^5cdot5^6=500000$.
answered 12 mins ago
Servaes
19.5k33686
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Here's a very unsophisticated approach: Let $n$ be the smallest such integer. Then there exist integers $a$ and $b$ such that $n=5a^5$ and $n=2b^2$. It follows that $a$ is a multiple of $2$, say $a=2a_1$, and $b$ is a multiple of $5$, say $b=5b_1$. Then
$$n=2^5cdot5cdot a_1^5qquadtext and qquad n=2cdot5^2cdot b_1^2.$$
This in turn shows that $a_1$ is a multiple of $5$, say $a_1=5a_2$, and $b_1$ is a multiple of $2$, say $b_1=2b_2$. Then
$$n=2^5cdot5^6cdot a_2^5qquadtext and qquad n=2^3cdot5^2cdot b_2^2.$$
This in turn shows that $b_2$ is a multiple of both $2$ and $5^2$, say $b_2=2cdot5^2cdot b_3$. Then
$$n=2^5cdot5^6cdot a_2^5qquadtext and qquad n=2^5cdot5^6cdot b_3^2.$$
This shows that $ngeq2^5cdot5^6$, and as you might expect a quick check shows that $n=2^5cdot5^6$ does indeed work, so $n=2^5cdot5^6=500000$.
answered 12 mins ago
Servaes
19.5k33686
Here's a very unsophisticated approach: Let $n$ be the smallest such integer. Then there exist integers $a$ and $b$ such that $n=5a^5$ and $n=2b^2$. It follows that $a$ is a multiple of $2$, say $a=2a_1$, and $b$ is a multiple of $5$, say $b=5b_1$. Then
$$n=2^5cdot5cdot a_1^5qquadtext and qquad n=2cdot5^2cdot b_1^2.$$
This in turn shows that $a_1$ is a multiple of $5$, say $a_1=5a_2$, and $b_1$ is a multiple of $2$, say $b_1=2b_2$. Then
$$n=2^5cdot5^6cdot a_2^5qquadtext and qquad n=2^3cdot5^2cdot b_2^2.$$
This in turn shows that $b_2$ is a multiple of both $2$ and $5^2$, say $b_2=2cdot5^2cdot b_3$. Then
$$n=2^5cdot5^6cdot a_2^5qquadtext and qquad n=2^5cdot5^6cdot b_3^2.$$
This shows that $ngeq2^5cdot5^6$, and as you might expect a quick check shows that $n=2^5cdot5^6$ does indeed work, so $n=2^5cdot5^6=500000$.
answered 12 mins ago
Servaes
19.5k33686
answered 12 mins ago
Servaes
19.5k33686
answered 12 mins ago
Servaes
19.5k33686
answered 12 mins ago
Servaes
19.5k33686
19.5k33686
add a comment |Â
add a comment |Â
J. DOEE is a new contributor. Be nice, and check out our Code of Conduct.
J. DOEE is a new contributor. Be nice, and check out our Code of Conduct.
J. DOEE is a new contributor. Be nice, and check out our Code of Conduct.
J. DOEE is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2976181%2fwhat-is-the-smallest-integer-greater-than-1-such-that-1-2-of-it-is-a-perfect-squ%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
3
It has to be divisible by $5$ and $2$ so you are looking for a multiple of $10$. Working with perfect fifths will get you through the numbers faster. The number $500000$ works so you can work down from there.
– John Douma
23 mins ago