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Modular Arithmetic Operations

Considering the remainder "modulo" an integer is a powerful, foundational tool in Number Theory. You already use in clocks and work modulo 12.

Level 1

         

What is the last digit when

\[ 1234 \times 5678 \]

is multiplied out?

If today is a Monday, then what day will it be 100 days later?

\[ \LARGE \color{red} 5^{\color{blue}8^{\color{green}{12}^{\color{purple}{15}^{\color{brown}{104}}}}} + \color{red}1\]

Determine the smallest prime divisor of the gigantic number above.

One of the seven goblets above is made of real gold. If you start counting at A and wind back and forth while counting (A, B, C, D, E, F, G, F, E, D, ...), then the golden goblet would be the \(1000^\text{th}\) one that you count.

Which one is the golden goblet?

If \(a\) is a negative odd number and \(b\) is a positive even number, then which of the following must be a positive even number?

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