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Smallest positive integer that is divisble by 2, 3 and 5 and that is also a perfect square and perfect cube...
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
edited Nov 20 '18 at 12:31
cansomeonehelpmeout
6,7483835
asked Nov 20 '18 at 12:12
mathpieuler
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.
add a comment |
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
edited Nov 20 '18 at 12:31
cansomeonehelpmeout
6,7483835
asked Nov 20 '18 at 12:12
mathpieuler
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.
$$(2cdot3cdot5)^{6m}$$
– 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
add a comment |
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
edited Nov 20 '18 at 12:31
cansomeonehelpmeout
6,7483835
asked Nov 20 '18 at 12:12
mathpieuler
13610
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
edited Nov 20 '18 at 12:31
cansomeonehelpmeout
6,7483835
asked Nov 20 '18 at 12:12
mathpieuler
13610
edited Nov 20 '18 at 12:31
cansomeonehelpmeout
6,7483835
asked Nov 20 '18 at 12:12
mathpieuler
13610
edited Nov 20 '18 at 12:31
cansomeonehelpmeout
6,7483835
edited Nov 20 '18 at 12:31
cansomeonehelpmeout
6,7483835
edited Nov 20 '18 at 12:31
cansomeonehelpmeout
6,7483835
6,7483835
asked Nov 20 '18 at 12:12
mathpieuler
13610
asked Nov 20 '18 at 12:12
mathpieuler
13610
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
add a comment |
$$(2cdot3cdot5)^{6m}$$
– 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
add a comment |
2 Answers
2
active
oldest
votes
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 $$(2cdot 3cdot 5)^6=30^6=729 000 000$$
answered Nov 20 '18 at 12:23
cansomeonehelpmeout
6,7483835
add a comment |
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 $30^6$.
answered Nov 20 '18 at 12:20
Akash Roy
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
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 $$(2cdot 3cdot 5)^6=30^6=729 000 000$$
answered Nov 20 '18 at 12:23
cansomeonehelpmeout
6,7483835
add a comment |
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 $$(2cdot 3cdot 5)^6=30^6=729 000 000$$
answered Nov 20 '18 at 12:23
cansomeonehelpmeout
6,7483835
add a comment |
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 $$(2cdot 3cdot 5)^6=30^6=729 000 000$$
answered Nov 20 '18 at 12:23
cansomeonehelpmeout
6,7483835
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 $$(2cdot 3cdot 5)^6=30^6=729 000 000$$
answered Nov 20 '18 at 12:23
cansomeonehelpmeout
6,7483835
answered Nov 20 '18 at 12:23
cansomeonehelpmeout
6,7483835
answered Nov 20 '18 at 12:23
cansomeonehelpmeout
6,7483835
answered Nov 20 '18 at 12:23
cansomeonehelpmeout
6,7483835
6,7483835
add a comment |
add a comment |
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 $30^6$.
answered Nov 20 '18 at 12:20
Akash Roy
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
add a comment |
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 $30^6$.
answered Nov 20 '18 at 12:20
Akash Roy
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
add a comment |
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 $30^6$.
answered Nov 20 '18 at 12:20
Akash Roy
1
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 $30^6$.
answered Nov 20 '18 at 12:20
Akash Roy
1
answered Nov 20 '18 at 12:20
Akash Roy
1
answered Nov 20 '18 at 12:20
Akash Roy
1
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
add a comment |
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
add a comment |
$$(2cdot3cdot5)^{6m}$$
– 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