if-the-lcm-and-gcf-of-three-numbers-are-360-and-6-other-numbers-are-18-and-60-Find-the-third-number-I-need-help-plz-

Question Number 72526 by liki last updated on 30/Oct/19

i f t h e l c m a n d g c f o f t h r e e n u m b e r s

a r e 360 a n d 6, o t h e r n u m b e r s a r e 18

a n d 60 . F i n d t h e t h i r d n u m b e r .

\dots I n e e d h e l p p l z \dots

Commented by TawaTawa last updated on 30/Oct/19

For three numbers

a \times b \times c = \frac{lcm (a, b, c) \times \gcd (a, b) \times \gcd (c, a)}{\gcd (a, b, c)}

Answered by mind is power last updated on 30/Oct/19

\gcd (a, b, c) = d

d ∣ a, b

d ∣ c

d \Rightarrow∣ \gcd (a, b), c \Rightarrow d ∣ \gcd ((\gcd (a, b), c)

\gcd ((\gcd (a, b), c) = d \Rightarrow d ∣ c

d ∣ \gcd (a, b) \Rightarrow d ∣ a & d ∣ b

\Rightarrow \gcd (a, b, c) = \gcd (\gcd (a, b), c)

lcm (a, b, c) = lcm (lcm (a, b), c))

m = lcm (a, b, c) \Rightarrow a ∣ m, b ∣ m \Rightarrow lcm (a, b) ∣ m

c ∣ m \Rightarrow lcm (lcm (a, b), c) ∣ m

d = lcm (lcm (a, b), c) \Rightarrow c ∣ d, lcm (a, b) ∣ d \Rightarrow a ∣ d, b ∣ d, c ∣ d \Rightarrow lcm (a, b, c) ∣ d

\Leftrightarrow lcm (lcm (a, b), c) = lcm (a, b, c)

after that a, b, c three number

lcm (a, b, c) = 360

\gcd (a, b, c) = 18

a = 6, b = 18

\Leftrightarrow {\begin{cases} lcm (6, 18, c) = 360 = lcm (lcm (18, 60), c) = 360 \\ \gcd (60, 18, c) = 6 = \gcd (\gcd (60, 18), c) = 6 \end{cases}

lcm (60, 18) = 6 lcm (10, 3) = 180, \gcd (60, 18) = 6

\Rightarrow lcm (180, c) = 360

\Rightarrow \gcd (6, c) = 6

c = 6 k \Rightarrow \gcd (6, 6 k) = 6 = 6 \gcd (k, 1) = 6

lcm (180, 6 k) = 360

6 lcm (30, k) = 360

lcm (30, k) = 60 = \frac{30 . k}{\gcd (30, k)} \Rightarrow 2 \gcd (30, k) = k

\gcd (30, k) \in {1, 2, 3, 5, 6, 10, 15, 30}

= 1 \Rightarrow k = 2 no

\gcd = 2 \Rightarrow k = 4 \Rightarrow

\gcd = 3 \Rightarrow k = 6 but \gcd (6, 30) = 6 no

\gcd = 5 \Rightarrow k = 10 \gcd (10, 30) = 10 no

\gcd = 6 \Rightarrow k = 12

\gcd = 10 \Rightarrow k = 20

\gcd = 15 \Rightarrow k = 30 non

\gcd = 30 \Rightarrow k = 60

k \in {4, 20, 60, 12}

c = 6 k

c \in {360, 120, 24, 72}

Commented by liki last updated on 30/Oct/19

t h a n k s s i r . .

Commented by mind is power last updated on 30/Oct/19

y^{'} re welcom

Commented by mr W last updated on 30/Oct/19

24 a n d 72 a l s o o k

Commented by mind is power last updated on 30/Oct/19

yes i fixed i took 6 but [the two number are 18, 60 not 18 and 6

lol

Answered by mr W last updated on 30/Oct/19

L C M = 360 = 2^{3} \times 3^{2} \times 5^{1}

G C D = 6 = 2^{1} \times 3^{1}

A = 18 = 2^{1} \times 3^{2}

B = 60 = 2^{2} \times 3^{1} \times 5^{1}

C = 2^{3} \times 3^{1 \dots 2} \times 5^{0 \dots 1}

= 8 \times 3 = 24

= 8 \times 3 \times 5 = 120

= 8 \times 9 = 72

= 8 \times 9 \times 5 = 360

Commented by TawaTawa last updated on 30/Oct/19

Sir help me check the question i solved in Q 72560

if i am correct

Commented by liki last updated on 31/Oct/19

∙ T h a n k y o u s i r

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