IQ Math - Olympiad Problem - Remainder - art of Solving Number Theory

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If γ€–24γ€—^x=γ€–243γ€—^y=72, find the value of ((x+5y)/xy). If γ€–24γ€—^x=γ€–243γ€—^y=72, find the value of ((x+5y)/xy).
Solve the equation : x^(x√x)=(x√x)^x. Solve the equation : x^(x√x)=(x√x)^x.
Find the value of (1-1/2^2 )(1-1/3^2 )(1-1/4^2 )(1-1/5^2 )……(1-1/γ€–2023γ€—^2 )(1-1/γ€–2024γ€—^2 ). Find the value of (1-1/2^2 )(1-1/3^2 )(1-1/4^2 )(1-1/5^2 )……(1-1/γ€–2023γ€—^2 )(1-1/γ€–2024γ€—^2 ).
Find the remainder when γ€–997γ€—^62-γ€–107γ€—^62 is divided by 23. Find the remainder when γ€–997γ€—^62-γ€–107γ€—^62 is divided by 23.
If a+b=2 and 1/a+1/b=2, find the value of a^2023+b^2023. If a+b=2 and 1/a+1/b=2, find the value of a^2023+b^2023.
If x=2+2^(1/3)+2^(2/3), find the value of x^3-6x^2+6x. If x=2+2^(1/3)+2^(2/3), find the value of x^3-6x^2+6x.
Find the smallest positive integer k such that γ€–(k-10)γ€—^4044 is greater thanγ€–2022γ€—^2022. Find the smallest positive integer k such that γ€–(k-10)γ€—^4044 is greater thanγ€–2022γ€—^2022.
If 2^28-1 is divided by 145, find the remainder. If 2^28-1 is divided by 145, find the remainder.