I want to ask one question

I was Thinking of this problem while my dinner any clue of how to find it??

Find the minimum value of $a$ for which the expression $a^{1024} \mod1024$ is maximum

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Brilliant Folks Could you help me???????????????????

Sumukh Bansal - 3 years, 6 months ago

@Otto Bretscher @Pi Han Goh @Anirudh Sreekumar @Brandon Monsen @Brian Charlesworth @Jon Haussmann Any Help??????????

Sumukh Bansal - 3 years, 6 months ago

There is no minimum value for $a$ .

Hint: Think of $a^{1024} \bmod{1024}$ and $(a-1024)^{1024} \bmod{1024}$ .

Pi Han Goh - 3 years, 6 months ago

Let's rephrase it. Find the minimum positive integral value of $a$ for which the expression $a^{1024} \mod1024$ is maximum The maximum value will be 1023 but how to find a

Sumukh Bansal - 3 years, 6 months ago

@Sumukh Bansal –

Hint: Why should we only look for odd $a$ 's?

Hint 2: Euler's totient function.

Hint 3: $a^{bc} \bmod D = (a^b \bmod D)^C \bmod D$ .

Pi Han Goh - 3 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$