A number theory problem by uvindu deeyamulla

Number Theory Level 1

Which of the following pairs of integers satisfy the equation 1 a + 1 b = 5 12 \dfrac1a + \dfrac1b =\dfrac5{12} ?

a = 3. b = 4 a=3.b=4 a = 2 , b = 6 a=2,b=6 a = 1 , b = 12 a=1,b=12 a = 6 , b = 4 a=6,b=4

Uvindu Deeyamulla by uvindu deeyamulla , 21, Sri Lanka

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2 solutions

Viki Zeta
Oct 9, 2016

1 a + 1 b = 5 12 ⟹ a + b a b = 5 12 ⟹ L C M ( a , b ) = 12 , such that a + b = 5 Factors of 12 = ± 1 , ± 2 , ± 3 , ± 4 , ± 6 , ± 12 ⟹ 6 + 4 6 × 4 = 10 24 = 5 12 , is one of possible positive solution \dfrac{1}{a} + \dfrac{1}{b} = \dfrac{5}{12} \\ \implies \dfrac{a+b}{ab} = \dfrac{5}{12} \\ \implies LCM(a, b) = 12, \text{ such that } a+b=5 \\ \text{Factors of 12 } = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \\ \boxed{\implies \dfrac{6 + 4}{6 \times 4} = \dfrac{10}{24} = \dfrac{5}{12} \text{, is one of possible positive solution}}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

a=4,b=6............................. 6 + 4 6 X 4 \frac{6+4}{6X4} = 10 24 \frac{10}{24} = 5 12 \frac{5}{12}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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