The LCM and the GCF of three intergers are 180 and 3 respectively. Two numbers are 45 and 60. What is the third number


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Question Number 41167 by Cheyboy last updated on 02/Aug/18

$T h e L C M a n d t h e G C F o f t h r e e$ $i n t e r g e r s a r e 180 a n d 3 r e s p e c t i v e l y .$ $T w o n u m b e r s a r e 45 a n d 60 .$ $W h a t i s t h e t h i r d n u m b e r$
	Answered by MJS last updated on 02/Aug/18
	$45=3^2 5 60=2^2 3×5 lcm(45, 60)=2^2 3^2 5=180 lcm(180, x)=180 ⇒ x=2^(0≤i≤2) 3^(0≤j≤2) 5^(0≤k≤1) gcf(45, 60)=3×5=15 gcf(15, x)=3 ⇒ x=2^(0≤i) 3^(1≤j) ⇒ x∈{3, 6, 9, 12, 18, 36}$
	$45 = 3^{2} 5$ $60 = 2^{2} 3 \times 5$ $lcm (45, 60) = 2^{2} 3^{2} 5 = 180$ $lcm (180, x) = 180$ $\Rightarrow x = 2^{0 ⩽ i ⩽ 2} 3^{0 ⩽ j ⩽ 2} 5^{0 ⩽ k ⩽ 1}$ $gcf (45, 60) = 3 \times 5 = 15$ $gcf (15, x) = 3$ $\Rightarrow x = 2^{0 ⩽ i} 3^{1 ⩽ j}$ $\Rightarrow x \in {3, 6, 9, 12, 18, 36}$
	Commented by Cheyboy last updated on 02/Aug/18

	$T h a n k y o u s i r . I t h o u g h i t w a s o n l y 3$
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