Question

What can be the last digit of a square of a natural number?

Answer 1

Answer:

Any of {0, 1, 4, 5, 6, 9}

Step-by-step explanation:

The squares of the digits 0–9 are ...

0, 1, 4, 9, 16, 25, 36, 49, 64, 91

The last digit of a square must be the last digit of one of these:

... {0, 1, 4, 5, 6, 9}

_____

Comment on the problem

You will see that the squares of 5 and 6 end in 5 and 6, respectively. Such a number is called an automorph. The last half of its square is the original number. These can be extended indefinitely, usually according to a simple rule. The 3-digit automorph ending in 6 is 376. (376² = 141,376). The 5-digit automorph ending in 5 is 90625. (90625² = 8212890625)

Automorphs can also be found in number bases other than base-10.

Answer 2

The last digit of the square of a natural number can be 0, 1, 4, 5, 6, or 9.

A natural number is a number resulting from a count of a finite number of things. Thus, integers from 0 are considered natural numbers, since they allow us to represent a certain amount of things.

To determine what can be the last digit of a square of a natural number, this number must be multiplied by itself:

• 1 x 1 = 1
• 2 x 2 = 4
• 3 x 3 = 9
• 4 x 4 = 16
• 5 x 5 = 25
• 6 x 6 = 36
• 7 x 7 = 49
• 8 x 8 = 64
• 9 x 9 = 81
• 10 x 10 = 100

Thus, the last digit of the square of a natural number can be 0, 1, 4, 5, 6, or 9.

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