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 April 9th, 2018, 08:19 AM #1 Newbie   Joined: May 2017 From: Moscow Posts: 6 Thanks: 0 System with gcd and lcm Can you help to solve this? $\displaystyle \begin{cases} a+b=120 \\ \langle a,b\rangle =6(a,b) \end{cases}$ Where $\displaystyle \langle a,b\rangle$ is lcm (a,b) gcd respectively. a,b are natural April 9th, 2018, 08:57 AM #2 Senior Member   Joined: Aug 2017 From: United Kingdom Posts: 312 Thanks: 111 Math Focus: Number Theory, Algebraic Geometry This isn't too hard if you recall $ab = \langle a,b \rangle (a,b)$. One way to solve it: Using the equality I just stated, $\langle a,b\rangle =6(a,b)$ is equivalent to $ab = 6(a,b)^2$. Now if we write $a = a' (a,b)$ and $b = b' (a,b)$, we get $a'b' (a,b)^2 = 6(a,b)^2$ so $a'b' = 6$. Case 1: $a' = 6, b' = 1$. In this case we have $a = 6(a,b)$ and $b = (a,b)$, so $a + b = 120$ implies $6(a,b) + (a,b) = 120$, i.e. $(a,b) = \frac{120}{7}$. This is impossible, so no solutions in this case. The case $b' = 6, a' = 1$ is dealt with the same way, with no solutions. Case 2: $a' = 2, b' = 3$. In this case, $a = 2(a,b)$ and $b = 3(a,b)$, so $a + b = 120$ becomes $2(a,b) + 3(a,b) = 120$, i.e. $(a,b) = \frac{120}{5} = 24$. This implies $a = 2 \times 24 = 48$, $b = 3 \times 24 = 72$. We check that this is indeed a solution: $a + b = 48 + 72 = 120$, and $ab = 48 \times 72 = 6 \times 24^2 = 6 \times (a,b)^2$ both hold. Similarly if $a' = 3, b' = 2$, we get the solution $a = 72, b = 48$. Thanks from idontknow April 9th, 2018, 09:00 AM #3 Senior Member   Joined: Dec 2015 From: somewhere Posts: 514 Thanks: 80 Just start with gcd(a,b) ▪ lcm(a,b) = ab Tags gcd, lcm, system Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post h80 Geometry 1 March 2nd, 2017 08:59 AM Jhenrique Geometry 1 August 29th, 2015 06:33 AM SGH Number Theory 6 August 30th, 2013 09:02 AM ARTjoMS Calculus 2 November 20th, 2011 04:28 AM zain Elementary Math 1 April 7th, 2008 02:23 PM

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