Given GCD of two numbers is 42 and their product is 15876. How many possible sets of members can be found?
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Given GCD of two numbers is 42 and their product is 15876. How many possible sets of numbers can be found?
I have no idea. I can only evaluate the lcm. Don't know how to get the answers.
elementary-number-theory
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up vote
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Given GCD of two numbers is 42 and their product is 15876. How many possible sets of numbers can be found?
I have no idea. I can only evaluate the lcm. Don't know how to get the answers.
elementary-number-theory
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Given GCD of two numbers is 42 and their product is 15876. How many possible sets of numbers can be found?
I have no idea. I can only evaluate the lcm. Don't know how to get the answers.
elementary-number-theory
Given GCD of two numbers is 42 and their product is 15876. How many possible sets of numbers can be found?
I have no idea. I can only evaluate the lcm. Don't know how to get the answers.
elementary-number-theory
asked Aug 22 at 9:18
user1942348
1,3501625
1,3501625
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2 Answers
2
active
oldest
votes
up vote
7
down vote
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You have two numbers that are divisible by $42$. We can write these as $42a$ and $42b$ for $a, b$ coprime (can you see why?). Their product is $15876$, so we have
$$
42^2ab = 15876 iff ab = 9.
$$
It it now enough to determine which $a, b$ satisfy $ab = 9$, where $a$ and $b$ are coprime.
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up vote
2
down vote
Use $gcd(x,y) times operatornamelcm(x,y) = xy$
beginarrayrrr
xy &= &15876 \
operatornamelcm(x,y) &= &378 \
hline
gcd(x,y) &= &42
endarray
Assume $x < y$. Use $gcd(p^a, p^b) = p^min(a,b)$ and
$operatornamelcm(p^a, p^b) = p^max(a,b)$ when $p$ is a prime number.
beginarrayrcr
gcd(x,y) &= &42 & 2^1 & 3^1 & 7^1 \
operatornamelcm(x,y) &= &378 & 2^1 & 3^3 & 7^1 \
xy &= &15876 & 2^2 & 3^4 & 7^2 \
hline
x &= &42 & 2^1 & 3^1 & 7^1 \
y &= &378 & 2^1 & 3^3 & 7^1 \
endarray
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
accepted
You have two numbers that are divisible by $42$. We can write these as $42a$ and $42b$ for $a, b$ coprime (can you see why?). Their product is $15876$, so we have
$$
42^2ab = 15876 iff ab = 9.
$$
It it now enough to determine which $a, b$ satisfy $ab = 9$, where $a$ and $b$ are coprime.
add a comment |Â
up vote
7
down vote
accepted
You have two numbers that are divisible by $42$. We can write these as $42a$ and $42b$ for $a, b$ coprime (can you see why?). Their product is $15876$, so we have
$$
42^2ab = 15876 iff ab = 9.
$$
It it now enough to determine which $a, b$ satisfy $ab = 9$, where $a$ and $b$ are coprime.
add a comment |Â
up vote
7
down vote
accepted
up vote
7
down vote
accepted
You have two numbers that are divisible by $42$. We can write these as $42a$ and $42b$ for $a, b$ coprime (can you see why?). Their product is $15876$, so we have
$$
42^2ab = 15876 iff ab = 9.
$$
It it now enough to determine which $a, b$ satisfy $ab = 9$, where $a$ and $b$ are coprime.
You have two numbers that are divisible by $42$. We can write these as $42a$ and $42b$ for $a, b$ coprime (can you see why?). Their product is $15876$, so we have
$$
42^2ab = 15876 iff ab = 9.
$$
It it now enough to determine which $a, b$ satisfy $ab = 9$, where $a$ and $b$ are coprime.
edited Aug 22 at 9:27
answered Aug 22 at 9:22
Bill Wallis
2,2361826
2,2361826
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add a comment |Â
up vote
2
down vote
Use $gcd(x,y) times operatornamelcm(x,y) = xy$
beginarrayrrr
xy &= &15876 \
operatornamelcm(x,y) &= &378 \
hline
gcd(x,y) &= &42
endarray
Assume $x < y$. Use $gcd(p^a, p^b) = p^min(a,b)$ and
$operatornamelcm(p^a, p^b) = p^max(a,b)$ when $p$ is a prime number.
beginarrayrcr
gcd(x,y) &= &42 & 2^1 & 3^1 & 7^1 \
operatornamelcm(x,y) &= &378 & 2^1 & 3^3 & 7^1 \
xy &= &15876 & 2^2 & 3^4 & 7^2 \
hline
x &= &42 & 2^1 & 3^1 & 7^1 \
y &= &378 & 2^1 & 3^3 & 7^1 \
endarray
add a comment |Â
up vote
2
down vote
Use $gcd(x,y) times operatornamelcm(x,y) = xy$
beginarrayrrr
xy &= &15876 \
operatornamelcm(x,y) &= &378 \
hline
gcd(x,y) &= &42
endarray
Assume $x < y$. Use $gcd(p^a, p^b) = p^min(a,b)$ and
$operatornamelcm(p^a, p^b) = p^max(a,b)$ when $p$ is a prime number.
beginarrayrcr
gcd(x,y) &= &42 & 2^1 & 3^1 & 7^1 \
operatornamelcm(x,y) &= &378 & 2^1 & 3^3 & 7^1 \
xy &= &15876 & 2^2 & 3^4 & 7^2 \
hline
x &= &42 & 2^1 & 3^1 & 7^1 \
y &= &378 & 2^1 & 3^3 & 7^1 \
endarray
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Use $gcd(x,y) times operatornamelcm(x,y) = xy$
beginarrayrrr
xy &= &15876 \
operatornamelcm(x,y) &= &378 \
hline
gcd(x,y) &= &42
endarray
Assume $x < y$. Use $gcd(p^a, p^b) = p^min(a,b)$ and
$operatornamelcm(p^a, p^b) = p^max(a,b)$ when $p$ is a prime number.
beginarrayrcr
gcd(x,y) &= &42 & 2^1 & 3^1 & 7^1 \
operatornamelcm(x,y) &= &378 & 2^1 & 3^3 & 7^1 \
xy &= &15876 & 2^2 & 3^4 & 7^2 \
hline
x &= &42 & 2^1 & 3^1 & 7^1 \
y &= &378 & 2^1 & 3^3 & 7^1 \
endarray
Use $gcd(x,y) times operatornamelcm(x,y) = xy$
beginarrayrrr
xy &= &15876 \
operatornamelcm(x,y) &= &378 \
hline
gcd(x,y) &= &42
endarray
Assume $x < y$. Use $gcd(p^a, p^b) = p^min(a,b)$ and
$operatornamelcm(p^a, p^b) = p^max(a,b)$ when $p$ is a prime number.
beginarrayrcr
gcd(x,y) &= &42 & 2^1 & 3^1 & 7^1 \
operatornamelcm(x,y) &= &378 & 2^1 & 3^3 & 7^1 \
xy &= &15876 & 2^2 & 3^4 & 7^2 \
hline
x &= &42 & 2^1 & 3^1 & 7^1 \
y &= &378 & 2^1 & 3^3 & 7^1 \
endarray
edited Aug 22 at 14:48
answered Aug 22 at 14:41
steven gregory
16.7k22155
16.7k22155
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