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# Number Theory

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The numbers \$24^2 = 576\$ and \$56^2 = 3136\$ are examples of perfect squares that have a units digits of \$6.\$
If the units digit of a perfect square is \$5,\$ then what are the possible values of the tens digit?

Jul 7, 2024

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The numbers \$24^2 = 576\$ and \$56^2 = 3136\$ are examples of perfect squares that have a units digits of \$6.\$
If the units digit of a perfect square is \$5,\$ then what are the possible values of the tens digit?

If a number ends with 5, its square will end with 25.

So the only possible value of the tens digit is 2.

How to square a number that ends with a 5.

Separate the 5 out of the number.

Change that 5 to 25.

Add 1 to the number that is left.

Multiply that by the original number.

Stick the 25 to the end of the product.

Examples:

152                              452                              752                              1252

1                  5              4                  5             7                  5              12                  5

1                25              4                25             7                25              12                25

1 x (1+1)    25              4 x (4+1)    25             7 x (7+1)    25              12 x (12+1)  25

1 x 2           25              4 x 5          25             7 x 8           25              12 x 13        25

225                              2025                           5625                             15625

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Jul 7, 2024