Multiplying factors of a positive number

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The task is to find a singel positive whole number(X), where all the factors of that number(X) multiplied together equal $24^240$




elementary-number-theory factoring




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  • Sorry but the question is not clear to me. The product of the factors of a number alwayf give that number. What am I missing?
    – gimusi
    Aug 29 at 9:10










  • Have you tried writing $24^240$ as a product of prime factors?
    – Bruce
    Aug 29 at 9:10











  • From my reading of the question you seem to be looking for a single positive integer where if you list all its factors and multiply them together the answer makes $24^240$ ... am I right?
    – Bruce
    Aug 29 at 9:12










  • Hi. We encourage the posters to show us their attempts so we can help them identify the difficulties. Please edit your question, otherwise the question would be put on hold or closed.
    – xbh
    Aug 29 at 9:30










  • Exactly, I have to find a singel positive number(X), where all the factors of that number(X) multiplied together equal 24^240
    – StillStudent
    Aug 29 at 9:32














up vote
-3
down vote

favorite












The task is to find a singel positive whole number(X), where all the factors of that number(X) multiplied together equal $24^240$




elementary-number-theory factoring




share|cite|improve this question






















  • Sorry but the question is not clear to me. The product of the factors of a number alwayf give that number. What am I missing?
    – gimusi
    Aug 29 at 9:10










  • Have you tried writing $24^240$ as a product of prime factors?
    – Bruce
    Aug 29 at 9:10











  • From my reading of the question you seem to be looking for a single positive integer where if you list all its factors and multiply them together the answer makes $24^240$ ... am I right?
    – Bruce
    Aug 29 at 9:12










  • Hi. We encourage the posters to show us their attempts so we can help them identify the difficulties. Please edit your question, otherwise the question would be put on hold or closed.
    – xbh
    Aug 29 at 9:30










  • Exactly, I have to find a singel positive number(X), where all the factors of that number(X) multiplied together equal 24^240
    – StillStudent
    Aug 29 at 9:32












up vote
-3
down vote

favorite









up vote
-3
down vote

favorite











The task is to find a singel positive whole number(X), where all the factors of that number(X) multiplied together equal $24^240$




elementary-number-theory factoring




share|cite|improve this question














The task is to find a singel positive whole number(X), where all the factors of that number(X) multiplied together equal $24^240$





elementary-number-theory factoring





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 29 at 9:55

























asked Aug 29 at 9:03









StillStudent

11




11











  • Sorry but the question is not clear to me. The product of the factors of a number alwayf give that number. What am I missing?
    – gimusi
    Aug 29 at 9:10










  • Have you tried writing $24^240$ as a product of prime factors?
    – Bruce
    Aug 29 at 9:10











  • From my reading of the question you seem to be looking for a single positive integer where if you list all its factors and multiply them together the answer makes $24^240$ ... am I right?
    – Bruce
    Aug 29 at 9:12










  • Hi. We encourage the posters to show us their attempts so we can help them identify the difficulties. Please edit your question, otherwise the question would be put on hold or closed.
    – xbh
    Aug 29 at 9:30










  • Exactly, I have to find a singel positive number(X), where all the factors of that number(X) multiplied together equal 24^240
    – StillStudent
    Aug 29 at 9:32
















  • Sorry but the question is not clear to me. The product of the factors of a number alwayf give that number. What am I missing?
    – gimusi
    Aug 29 at 9:10










  • Have you tried writing $24^240$ as a product of prime factors?
    – Bruce
    Aug 29 at 9:10











  • From my reading of the question you seem to be looking for a single positive integer where if you list all its factors and multiply them together the answer makes $24^240$ ... am I right?
    – Bruce
    Aug 29 at 9:12










  • Hi. We encourage the posters to show us their attempts so we can help them identify the difficulties. Please edit your question, otherwise the question would be put on hold or closed.
    – xbh
    Aug 29 at 9:30










  • Exactly, I have to find a singel positive number(X), where all the factors of that number(X) multiplied together equal 24^240
    – StillStudent
    Aug 29 at 9:32















Sorry but the question is not clear to me. The product of the factors of a number alwayf give that number. What am I missing?
– gimusi
Aug 29 at 9:10




Sorry but the question is not clear to me. The product of the factors of a number alwayf give that number. What am I missing?
– gimusi
Aug 29 at 9:10












Have you tried writing $24^240$ as a product of prime factors?
– Bruce
Aug 29 at 9:10





Have you tried writing $24^240$ as a product of prime factors?
– Bruce
Aug 29 at 9:10













From my reading of the question you seem to be looking for a single positive integer where if you list all its factors and multiply them together the answer makes $24^240$ ... am I right?
– Bruce
Aug 29 at 9:12




From my reading of the question you seem to be looking for a single positive integer where if you list all its factors and multiply them together the answer makes $24^240$ ... am I right?
– Bruce
Aug 29 at 9:12












Hi. We encourage the posters to show us their attempts so we can help them identify the difficulties. Please edit your question, otherwise the question would be put on hold or closed.
– xbh
Aug 29 at 9:30




Hi. We encourage the posters to show us their attempts so we can help them identify the difficulties. Please edit your question, otherwise the question would be put on hold or closed.
– xbh
Aug 29 at 9:30












Exactly, I have to find a singel positive number(X), where all the factors of that number(X) multiplied together equal 24^240
– StillStudent
Aug 29 at 9:32




Exactly, I have to find a singel positive number(X), where all the factors of that number(X) multiplied together equal 24^240
– StillStudent
Aug 29 at 9:32










1 Answer
1






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2
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First we write $24^240$ as the product of its prime factors:
$$24^240=2^7203^240$$
Then note that for any positive integer $n=prod_1=1^m p_i^alpha_i$, where each $p_i$ denotes a distinct prime number, the product of all its factors are
$$prod_i_1=0^alpha_1prod_i_2=0^alpha_2cdotsprod_i_m=0^alpha_mp_1^i_1p_2^i_2ldots p_m^i_m$$



From this information we can infer that such number $n$ has the form $2^a3^b$ for some positive integer $a,b$. Then multiplying all factors together, we have
$$prod_i=0^aprod_j=0^b2^i3^j=2^(b+1)(1+2+ldots+a)3^(a+1)(1+2+ldots+b)=2^7203^240$$
(exercise: show that the expressions on the left and middle of the equation are equal)



Now we have two equations and two unknowns in the exponents of $2$ and $3$. Can you finish the rest?






share|cite|improve this answer






















  • +1 Nice solution :)
    – gandalf61
    Aug 29 at 10:06










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1 Answer
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active

oldest

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes








up vote
2
down vote













First we write $24^240$ as the product of its prime factors:
$$24^240=2^7203^240$$
Then note that for any positive integer $n=prod_1=1^m p_i^alpha_i$, where each $p_i$ denotes a distinct prime number, the product of all its factors are
$$prod_i_1=0^alpha_1prod_i_2=0^alpha_2cdotsprod_i_m=0^alpha_mp_1^i_1p_2^i_2ldots p_m^i_m$$



From this information we can infer that such number $n$ has the form $2^a3^b$ for some positive integer $a,b$. Then multiplying all factors together, we have
$$prod_i=0^aprod_j=0^b2^i3^j=2^(b+1)(1+2+ldots+a)3^(a+1)(1+2+ldots+b)=2^7203^240$$
(exercise: show that the expressions on the left and middle of the equation are equal)



Now we have two equations and two unknowns in the exponents of $2$ and $3$. Can you finish the rest?






share|cite|improve this answer






















  • +1 Nice solution :)
    – gandalf61
    Aug 29 at 10:06














up vote
2
down vote













First we write $24^240$ as the product of its prime factors:
$$24^240=2^7203^240$$
Then note that for any positive integer $n=prod_1=1^m p_i^alpha_i$, where each $p_i$ denotes a distinct prime number, the product of all its factors are
$$prod_i_1=0^alpha_1prod_i_2=0^alpha_2cdotsprod_i_m=0^alpha_mp_1^i_1p_2^i_2ldots p_m^i_m$$



From this information we can infer that such number $n$ has the form $2^a3^b$ for some positive integer $a,b$. Then multiplying all factors together, we have
$$prod_i=0^aprod_j=0^b2^i3^j=2^(b+1)(1+2+ldots+a)3^(a+1)(1+2+ldots+b)=2^7203^240$$
(exercise: show that the expressions on the left and middle of the equation are equal)



Now we have two equations and two unknowns in the exponents of $2$ and $3$. Can you finish the rest?






share|cite|improve this answer






















  • +1 Nice solution :)
    – gandalf61
    Aug 29 at 10:06












up vote
2
down vote










up vote
2
down vote









First we write $24^240$ as the product of its prime factors:
$$24^240=2^7203^240$$
Then note that for any positive integer $n=prod_1=1^m p_i^alpha_i$, where each $p_i$ denotes a distinct prime number, the product of all its factors are
$$prod_i_1=0^alpha_1prod_i_2=0^alpha_2cdotsprod_i_m=0^alpha_mp_1^i_1p_2^i_2ldots p_m^i_m$$



From this information we can infer that such number $n$ has the form $2^a3^b$ for some positive integer $a,b$. Then multiplying all factors together, we have
$$prod_i=0^aprod_j=0^b2^i3^j=2^(b+1)(1+2+ldots+a)3^(a+1)(1+2+ldots+b)=2^7203^240$$
(exercise: show that the expressions on the left and middle of the equation are equal)



Now we have two equations and two unknowns in the exponents of $2$ and $3$. Can you finish the rest?






share|cite|improve this answer














First we write $24^240$ as the product of its prime factors:
$$24^240=2^7203^240$$
Then note that for any positive integer $n=prod_1=1^m p_i^alpha_i$, where each $p_i$ denotes a distinct prime number, the product of all its factors are
$$prod_i_1=0^alpha_1prod_i_2=0^alpha_2cdotsprod_i_m=0^alpha_mp_1^i_1p_2^i_2ldots p_m^i_m$$



From this information we can infer that such number $n$ has the form $2^a3^b$ for some positive integer $a,b$. Then multiplying all factors together, we have
$$prod_i=0^aprod_j=0^b2^i3^j=2^(b+1)(1+2+ldots+a)3^(a+1)(1+2+ldots+b)=2^7203^240$$
(exercise: show that the expressions on the left and middle of the equation are equal)



Now we have two equations and two unknowns in the exponents of $2$ and $3$. Can you finish the rest?







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Aug 29 at 9:46

























answered Aug 29 at 9:38









Alvin

564




564











  • +1 Nice solution :)
    – gandalf61
    Aug 29 at 10:06
















  • +1 Nice solution :)
    – gandalf61
    Aug 29 at 10:06















+1 Nice solution :)
– gandalf61
Aug 29 at 10:06




+1 Nice solution :)
– gandalf61
Aug 29 at 10:06

















 

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