Question

if the unit digit of a perfect square is 1 then the unit digit of its square root is ​

Answer 1

Answer:

1 or 9

Method used:

Modulo congruence

Step-by-step explanation:

Let a perfect square be x², whose unit digit is 1. As x² is a perfect square, x is an integer.

x² can be the following,

1, 81, 121, 361, 441, 841, 961, 1521,...

It's always true that every numbers leave their unit digits as remainder on division by 10. Here, as the unit digit of x² is 1, so x² leaves remainder 1 on division by 10.

According to modulo congruence this can be written as the following:

x² ≡ 1 (mod 10)

We can take the square roots, then it will be,

√x² ≡ √1 (mod 10)

⇒ x ≡ ±1 (mod 10)

Now we get two instances that if x² leaves remainder 1 on division by 10, or the unit digit of x² is 1, then x ≡ ±1 (mod 10), which means x leaves remainder either 1 or -1 on division by 10.

From each, we can find the unit digit of x, which is the answer.

Consider x ≡ 1 (mod 10). Here we can already get a clear image that x leaves remainder 1 on division by 10.

[∵ If a ≡ b (mod m), then 'a' leaves remainder 'b' on division by 'm'.]

Thus, unit digit of x is 1.

Consider x ≡ -1 (mod 10). This means that x leaves remainder -1 on division by 10. But the remainder -1 is not a positive integer, and also -1 can't be the unit digit of x.

Here the fact given below is made use of:

"a ≡ b (mod m)  if and only if  a + m ≡ b (mod m)"

So we have to add 10 to both the sides x and -1. Thus we get,

x + 10 ≡ -1 + 10 (mod 10)

x + 10 ≡ 9 (mod 10)

According to the above fact, as x + 10 ≡ 9 (mod 10), then x ≡ 9 (mod 10).

This means x leaves remainder 9 on division by 10.

Thus, unit digit of x is 9.

So the answers are 1 or 9, which can also be written as 5 ± 4.