Euler Totient Function Equations (Problem 2 2 )

Number Theory Level 2

ϕ ( x ) = ϕ ( x + 3 ) = ϕ ( 2 x ) = ϕ ( 2 x − 2 ) = 4 \phi(x) = \phi(x + 3) = \phi(2x) = \phi(2x - 2) = 4

Find x x satisfying the equation above.

Notation: ϕ ( ⋅ ) \phi(\cdot) denotes the Euler's totient function .


The answer is 5.

Yajat Shamji by Yajat Shamji , 16, United Kingdom

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1 solution

Possible solutions for ϕ ( x k ) = 4 \phi(x_k) = 4 are as follows:

x 1 × 1 2 = 4 ⟹ x 1 = 8 x 2 × 4 5 = 4 ⟹ x 2 = 5 x 3 × 1 2 × 4 5 = 4 ⟹ x 3 = 10 \begin{array} {ll} x_1 \times \dfrac 12 = 4 & \implies x_1 = 8 \\ x_2 \times \dfrac 45 = 4 & \implies x_2 = 5 \\ x_3 \times \dfrac 12 \times \dfrac 45 = 4 & \implies x_3 = 10 \end{array}

Therefore x = x 2 = 5 x=x_2 = \boxed 5 , x 1 = 8 = x + 3 x_1 = 8 = x+3 , x 3 = 10 = 2 x x_3 = 10 = 2x , and 2 x − 2 = 8 = x 1 2x-2 = 8 = x_1 .

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Great solution!

Yajat Shamji - 10 months, 1 week ago

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Glad that you like it again.

Chew-Seong Cheong - 10 months, 1 week ago

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