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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 3-4, Math Kangaroo 5-6, Math Kangaroo 7-8.

Cryptarithms are arithmetic puzzles.

Most of the times, cryptarithms consist of presenting an arithmetic operation where some of the digits are not shown. Often, they hidden digits are replaced by a symbol such as '?' or '*'.

Example in which the asterisk is a placeholder for hidden digits

In this example we can either immediately notice that:

- the only way to reproduce the '2' at the beginning of the number and one place value up is to multiply it by 10;
- the only way to reproduce the '3' as the last digit is to multiply it by a 1;

Therefore the operation is:

Other cryptarithms replace hidden digits by letters and each letter represents a specific digit.

Example in which letters are placeholders for digits:

Expanding the numbers in base 10:

From here we see that:

In order to obtain a zero for the tens digit, b must be 5.

Therefore:

and the operation is:

Useful facts for solving cryptarithms:

- two digits add up to a maximum of 18
- by adding two digits we carry over at most a 1

Cryptarithms are used by the examiner to verify good logical thinking.