Find more such patterns by observing the numbers and their squares from the table you filled earlier.

Question

Homi · Accepted Answer

We can find interesting patterns by looking at numbers and their squares.

Step 1 — Pattern of zeros in squares

Let us observe the numbers that end with zeros and their squares from the given diagram.

If a number has one zero at the end, like 10, 20, or 40, its square will have two zeros. $10^2 = 100$ $20^2 = 400$ $40^2 = 1600$

If a number has two zeros at the end, like 100, 200, 700, or 900, its square will have four zeros. $100^2 = 10000$ $200^2 = 40000$ $700^2 = 490000$ $900^2 = 810000$

$\boxed{\text{The number of zeros in a square is double the number of zeros in the original number.}}$

Diagram 1

Step 2 — Patterns of units digits in squares

Let us observe the units digit of numbers and their squares from a table of squares.

If a number ends with 1 or 9, its square ends with 1. $1^2 = 1$ $9^2 = 81$ $11^2 = 121$ $19^2 = 361$

If a number ends with 2 or 8, its square ends with 4. $2^2 = 4$ $8^2 = 64$ $12^2 = 144$ $18^2 = 324$

If a number ends with 3 or 7, its square ends with 9. $3^2 = 9$ $7^2 = 49$ $13^2 = 169$ $17^2 = 289$

If a number ends with 4 or 6, its square ends with 6. $4^2 = 16$ $6^2 = 36$ $14^2 = 196$ $16^2 = 256$

If a number ends with 5, its square ends with 5. $5^2 = 25$ $15^2 = 225$

If a number ends with 0, its square ends with 00. $10^2 = 100$ $20^2 = 400$

Answer

(i) If a number has a 4 or 6 in the units place, its square ends in 6. (ii) If a number has a 3 or 7 in the units place, its square ends in 9. (iii) If a number has a 2 or 8 in the units place, its square ends in 4. (iv) If a number has a 1 or 9 in the units place, its square ends in 1. (v) If a number has a 5 in the units place, its square ends in 5. (vi) If a number has a 0 in the units place, its square ends in 00.

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