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# Mathematical Fundamentals

The essential tools for algebra, geometry, probability, and logic!

\[ \Large \begin{array} { c c c } & 1 & \color{orange}{B} \\ + & \color{orange}{B} & 6 \\ \hline & 7 & 1 \\ \end{array} \]

What digit in place of \(\color{orange}{B}\) would make this sum true?

\[\large \begin{array} { l l l l }

& \color{green}{Z} & \color{green}{Z} \\
& \color{green}{Z} & \color{green}{Z} \\
& & \color{green}{Z} \\
& & \color{green}{Z} \\
+ & &\color{green}{Z} \\
\hline 1 & 0 & 0
\end{array} \]

What digit is \(\color{green}{Z}\)?

\[ \Large \begin{array} {c c c c } & & \color{red}{A} & \color{blue}{B} \\ + & & \color{red}{A} & \color{blue}{B} \\ \hline & 1 & 3 & 8 \\ \end{array} \]

What is the value of \(\color{red}{A} \color{#333333}{?}\)

\[\Large \begin{array} {c c c } & 1 & \color{blue}{E} \\ \times & & \color{blue}{E} \\ \hline & 9 & \color{blue}{E} \\ \end{array} \]

What digit in place of \(\color{blue}{E}\) would make this multiplication true?

**Cryptograms** are puzzles where capital letters stand in for the digits of a number. If the same letter is used twice, it’s the same digit in both places. And, in all of our puzzles, different letters represent different digits.

Solve enough cryptograms, and you’ll start to see patterns--tricks that will help you solve puzzles like these faster and faster. There are patterns in all numbers. For instance, one pattern is that any number greater than \(0\) that ends in \(0\) is divisible by \(10\). \(30, 150, 4000,\) and \(456827382930\) are all divisible by 10. You can tell just by seeing that \(0\) at the end.

Other patterns are trickier, but can also be more useful. For example, all of the numbers below are divisible by 9, but what else do their digits all have in common? \[ \color{orange} 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117,...\]

\[\Large \begin{array} {c c c } & \color{red}{D} & \color{green}{T} \\ \times & & 9 \\ \hline \\ \color{red}{D} & \color{red}{D} & 1 \\ \end{array} \]

What is the value of \(\color{red}{D} \color{#333333}{?}\)

Strategy Tip: Use the fact that \(\color{red}{DD} \color{#333333}{1}\) is divisible by 9. Also, consider that the pattern in the digits of multiples of 9 observed on the previous page comes from the fact that

each 100 in a number = eleven 9s + 1 leftover;

each 10 in a number = 9 + 1 leftover.

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