When a number multiplied by itself is called the number square. If a number having all 1’s is squared then the middle number in the square will be the number of one’s being multiplied. Get the definition of the multiplied by itself, facts, properties, and solved example questions in the following sections.
Also, Refer:
What is a Number Multiplied by Itself?
Multiplying a number with the same number is called the square of that number. The square of a number is represented with power 2 or superscript 2. If 1 multiplied by itself gives 1. Likewise, if 11 multiplied by 11 gives 121 it says that the middle number is 2 which is nothing but the number of 1’s being multiplied.
Examples:
12 x 12 = 144
111 x 111 = 12321
Properties of Squares
- A number that is having 2, 3, 7, or 8 at its end place can never be a perfect square number.
- The number of zeros at the end of a number determines if it is a perfect square or not. If a number ends with an even number of zeros then it can be a perfect square but if a number ends with an odd number of zeros then it can’t be a perfect square.
- The Square of an even number is an even number and the square of an odd number is an odd number.
- The square of a number n is equal to the sum of the first n odd natural numbers.
Multiplied a Number by Itself Examples
Question 1:
Find the product and write the middle number.
(a) 111 x 111
(b) 1111 x 1111
(c) 11111 x 11111
Solution:
(a) Given that,
111 x 111
111 x 111 = 12321
The middle number in the square is 3.
(b) Given that,
1111 x 1111
1111 x 1111 = 1234321
The middle number in the square is 4.
(c) Given that,
11111 x 11111
11111 x 11111 = 123454321
The middle number in the square is 5.
By observing these solutions, we can say that when a number having all 1’s multiplied by itself, then the product will have the number of 1’s in the middle and the remaining numbers are 1, 2, 3 in reverse order on both sides.
Question 2:
Solve the following
(a) 111111 x 111111
(b) 11 x 11
Solution:
(a) Given that,
111111 x 111111
It is a 6 digit number having all 1’s in it. So, the product has 6 in the middle and 12345 in reverse order.
Therefore, 111111 x 111111 = 12345654321
(b) Given that,
11 x 11
It is a 2 digit number having all 1’s in it. So, the product has 2 in the middle and 1 in reverse order.
Therefore, 11 x 11 = 121
Question 3:
Find the product of the following
(a) 57 x 57
(b) 88 x 88
(c) 62 x 62
Solution:
(a) Given that,
57 x 57
Find the square of 57 using the math trick.
Add the unit digit of the number to the original number.
7 + 57 = 64
Multiply the sum by the ten’s digit of the original number.
64 x 5 = 320
Find the square of the last digit of the original number.
7² = 49
Append the obtained numbers in the above two steps.
3200 + 49 = 3249
Therefore, 57 x57 = 3249
(b) Given that,
88 x 88
Find the square of 88 using the math trick.
Add the unit digit of the number to the original number.
8 + 88 = 96
Multiply the sum by the ten’s digit of the original number.
96 x 8 = 768
Find the square of the last digit of the original number.
8² = 64
Append the obtained numbers in the above two steps.
7680 + 64 = 7744
Therefore, 88 x 88 = 7744
(c) Given that,
62 x 62
Find the square of 62 using the math trick.
Add the unit digit of the number to the original number.
2 + 62 = 64
Multiply the sum by the ten’s digit of the original number.
64 x 6 = 384
Find the square of the last digit of the original number.
2² = 4
Append the obtained numbers in the above two steps.
3844
Therefore, 62 x 62 = 3844
FAQs on Number Multiplied by Itself
1. What happens when you multiply a number by itself?
In mathematics, a number being multiplied by itself is called the square of that number. We call the result of the square of a whole number is a square or a perfect square. A perfect square is any number that can be a whole number raised to power 2.
2. What is the result of a three-digit number having all 1’s multiplied by itself?
A there digit number having all 1’s is 111. When 111 is multiplied by itself the result will have 3 in the middle because it is a 3 digit number and the remaining numbers are 1, 2. So, 111 x 111 = 12321.
3. When a number is multiplied by itself twice?
A number multiplied by itself two times is called the number square. An example is 5 x 5 = 25 here 25 is the square of 5.