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if-4a-7b-2c-and-dcm-a-b-c-3-find-a-b-c-




Question Number 146470 by mathdanisur last updated on 13/Jul/21
if   4a=7b=2c  and   dcm(a;b;c)=3  find   a+b+c=?
$${if}\:\:\:\mathrm{4}{a}=\mathrm{7}{b}=\mathrm{2}{c}\:\:{and}\:\:\:\boldsymbol{{dcm}}\left({a};{b};{c}\right)=\mathrm{3} \\ $$$${find}\:\:\:{a}+{b}+{c}=? \\ $$
Answered by Rasheed.Sindhi last updated on 13/Jul/21
if   4a=7b=2c  and   gcd(a,b,c)=3  a+b+c=?  a=(1/2)c,b=(2/7)c  ∵ a,b,c ∈N  ∴2 ∣ c & 7 ∣ c  Also gcd(a,b,c)=3  ∴ 3 ∣ c     c=2×3×7=42    a=21     b=12  a+b+c=21+12+42=75  Verification:  4(21)=7(12)=2(42)=84  gcd(21,12,42)=3
$${if}\:\:\:\mathrm{4}{a}=\mathrm{7}{b}=\mathrm{2}{c}\:\:{and}\:\:\:\boldsymbol{{gcd}}\left({a},{b},{c}\right)=\mathrm{3} \\ $$$${a}+{b}+{c}=? \\ $$$${a}=\frac{\mathrm{1}}{\mathrm{2}}{c},{b}=\frac{\mathrm{2}}{\mathrm{7}}{c} \\ $$$$\because\:{a},{b},{c}\:\in\mathbb{N} \\ $$$$\therefore\mathrm{2}\:\mid\:{c}\:\&\:\mathrm{7}\:\mid\:{c} \\ $$$${Also}\:\boldsymbol{{gcd}}\left({a},{b},{c}\right)=\mathrm{3} \\ $$$$\therefore\:\mathrm{3}\:\mid\:{c} \\ $$$$\:\:\:{c}=\mathrm{2}×\mathrm{3}×\mathrm{7}=\mathrm{42} \\ $$$$\:\:{a}=\mathrm{21} \\ $$$$\:\:\:{b}=\mathrm{12} \\ $$$${a}+{b}+{c}=\mathrm{21}+\mathrm{12}+\mathrm{42}=\mathrm{75} \\ $$$${Verification}: \\ $$$$\mathrm{4}\left(\mathrm{21}\right)=\mathrm{7}\left(\mathrm{12}\right)=\mathrm{2}\left(\mathrm{42}\right)=\mathrm{84} \\ $$$$\boldsymbol{{gcd}}\left(\mathrm{21},\mathrm{12},\mathrm{42}\right)=\mathrm{3} \\ $$
Commented by mathdanisur last updated on 13/Jul/21
thank you Ser cool
$${thank}\:{you}\:{Ser}\:{cool} \\ $$

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