A number theory problem by Megha Malyala

Number Theory Level 1

1 6 ! + 1 7 ! = x 8 ! \large \frac{1}{6!}+\frac{1}{7!}=\frac{x}{8!}

Find x x .

25 64 36 16 81

Megha Malyala by Megha Malyala , 30, India

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2 solutions

Áron Bán-Szabó
Jul 26, 2017

1 6 ! + 1 7 ! = 6 ! + 7 ! 6 ! ∗ 7 ! = x 8 ! ⇒ x ( 6 ! ∗ 7 ! ) = 8 ! ∗ ( 6 ! + 7 ! ) = 8 ! ∗ 6 ! + 8 ! ∗ 7 ! 7 ! ( 6 ! ∗ x ) = 7 ! ( 8 ∗ 6 ! + 8 ! ) ⇒ 6 ! ∗ x = 8 ∗ 6 ! + 8 ! 6 ! ( x ) = 6 ! ( 8 + 7 ∗ 8 ) ⇒ x = 8 + 56 = 64 \begin{aligned} \dfrac{1}{6!}+\dfrac{1}{7!} & = \dfrac{6!+7!}{6!*7!} \\ & = \dfrac{x}{8!} \\ \ \\ \Rightarrow x(6!*7!) & = 8!*(6!+7!) \\ & = 8!*6!+8!*7! \\ 7!(6!*x) & = 7!(8*6!+8!) \\ \Rightarrow 6!*x & = 8*6!+8! \\ 6!(x) & = 6!(8+7*8) \\ \ \\ \Rightarrow x & = 8+56=\boxed{64} \end{aligned}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

IMO, it is often better to use L C M ( 6 ! , 7 ! ) LCM ( 6!, 7! ) as the denominator.

Calvin Lin Staff - 3 years, 10 months ago

Hey it's wrong

Kanha Sahu - 3 years, 9 months ago
Chew-Seong Cheong
Jul 27, 2017

x 8 ! = 1 6 ! + 1 7 ! x 8 ! = 7 × 8 8 ! + 8 8 ! ⟹ x = 56 + 8 = 64 \begin{aligned} \frac x{8!} & = \frac 1{6!} + \frac 1{7!} \\ \frac x{8!} & = \frac {7\times 8}{8!} + \frac 8{8!} \\ \implies x & = 56 + 8 = \boxed{64} \end{aligned}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

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