Then Sherlock said, "Elementary"

Number Theory Level 3

Sherlock is learning modular arithmetic. He divides $4^{99}$ by $17$ and finds the residue. What is the integer that he finds?

This problem is a part of the set Along Came A Spider.

The answer is 13.

2 solutions

Sheikh Sakib Ishrak Shoumo
Dec 18, 2014

$4^{99} \equiv (4^2)^{49} . 4 \equiv (-1)^{49} . 4 \equiv (-4)\equiv 13\; ( \; mod\; 17 )$

Thus, the answer is $\boxed{13}$

how did -1 come to be??

Yuki Kuriyama - 5 years, 9 months ago

Why is this question Flagged?

Mehul Arora - 6 years, 5 months ago

The problem was previously flagged because it was initially unclear. It has since been cleared up by the creator, and I have unflagged it.

The problem previously stated:

It took Sherlock 99 days to solve a case. Everyday, he used only one vehicle, the options being : bicycle, motorbike, gyro-copter and a Lamborghini. Now, the criminal, who now lives in the jail, jots down every possible combinations of vehicles Sherlock could use (that is an arduous task). After that, he starts to send 17 of the combinations each to his friends via some arcane medium. Obviously, he finds out that the last voucher did not contain 17 combinations at all. But Sherlock knows the trick, and so, just monitoring the activities of the jailed criminal a while, he tells us the number of combinations contained in the last voucher. What is the number, however?

Calvin Lin Staff - 6 years, 5 months ago

All right Sir!! I see... Thank You...

P.S. You are the Best!!

Mehul Arora - 6 years, 5 months ago

Juan Cruz Roldán
Dec 1, 2020

$4^5 \equiv 4$

$4^6 \equiv 16$

$4^7 \equiv 13$

$4^8 \equiv 1$

$99 \equiv 3 \equiv 7 \ \mod (4)$

$\therefore 4^{99} \equiv 13 \mod (17)$

Then Sherlock said, "Elementary"

The answer is 13.

2 solutions

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