I am just messing

Number Theory Level 2

The value of { 5 2 n 24 } \{\frac{5^{2n}}{24}\} , n ∈ N n\in \mathbb {N} where { . } \{.\} denotes fractional part of x x is

5 24 \frac{5}{24} 1 24 \frac{1}{24} none 0 1 5 \frac{1}{5}

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Tanishq Varshney by Tanishq Varshney , 23, India

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1 solution

5 2 n 24 = 2 5 n 24 = ( 24 + 1 ) n 24 ⇒ 5 2 n ≡ 1 ( m o d 24 ) \dfrac {5^{2n}}{24}=\dfrac {25^{n}}{24} = \dfrac {(24+1)^{n}}{24}\quad \Rightarrow 5^{2n} \equiv 1 \pmod{24}

⇒ { 5 2 n 24 } = 5 2 n 24 − ⌊ 5 2 n 24 ⌋ = 1 24 \Rightarrow \left \{ \dfrac {5^{2n}}{24} \right \} = \dfrac {5^{2n}}{24} - \left \lfloor \dfrac {5^{2n}}{24} \right \rfloor = \boxed {\dfrac {1}{24}}

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sir shouldn't it be ( 24 + 1 ) n 24 \frac{(24+1)^{n}}{24}

Tanishq Varshney - 6 years, 2 months ago

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Yes, pardon this old man

Chew-Seong Cheong - 6 years, 2 months ago

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