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# Basic Applications of Modular Arithmetic

Solve integer equations, determine remainders of powers, and much more with the power of Modular Arithmetic.

On the first day of class, a teacher writes his name on the board in a secret code:

NS. TNJUI

What is the actual last letter of his last name?

DWWDFN DW GDZQ.

Which letter from the original message does the Q in the encoded message represent?

**3** Caesar Shifts: first a \(+11\) Caesar Shift, then a \(+24\) Caesar Shift, then finally a \(+17\) Caesar Shift. How will the spy encode the letter X?

DRKYVDRKZTJ ZJ KYV HLVVE FW KYV JTZVETVJ, REU ELDSVI KYVFIP ZJ KYV HLVVE FW DRKYVDRKZTJ.

The message above was encoded using a Caesar Shift. The letter pair "KY" appears frequently in the encoded message. Which of these common letter pairs could the encoded letter pair "KY" represent?

Which of these Caesar Shifts encodes the letter A differently than the others?

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