Question

A number with 100 digits is given. The first two digits of that number are 2 and 9.
How many digits has a square of that number?​

Answer 1

Step-by-step explanation:

Given:-

A number with 100 digits is given. The first two digits of that number are 2 and 9.

To find:-

How many digits has a square of that number?

Solution:-

Number of digits in the given number = 100

The first two digits in the number = 2 and 9

Wr know that

The number of digits in the square of a number with one digit (from 1-9) = 1 or 2

since 1^2=1 and 9^2=81

The number of digits in the square of a number with two digits (from 10-99)= 3 or 4

Since 10^2 = 100 and 99^2 = 9801

The number of digits in the square of a number with three digits (from 100-999)= 4 or 5

If we continue like this we get the following general formula:

1 or 2 can be written as 2(1)-1 or 2(1)

3 or 4 can be written as 2(2)-1 or 2(2)

5 or 6 can be written as 2(3)-1 or 2(3)

.

.

.

The number of digits in the square of a number with n digits =2n-1 or 2n

So ,

The number of digits in the square of a number with 100 digits =

2(100)-1 or 2(100)

=>200-1 or 200

=>199 or 200

If it has 2 in its units place it has 199 digits in the square of the number.

If it has 9 in its units place it has 200 digits in the square of the number.

Answer:-

The number of digits in the square of a number with 100 digits = 199 or 200

1)If it has 2 in its units place it has 199 digits in the square of the number.

(The first two digits like 29)

2)If it has 9 in its units place it has 200 digits in the square of the number.

(The first two digits like 92)

Used formulae:-

• The number of digits in the square of a number with n digits =2n-1 or 2n

Answer 2

Given : a number   with 100 digits  

The first two digits of that number are 2 and 9

To Find : How many digits has a square of that number

29yyyy________

Hence its less than

300000_________

3 with 99  0's

and greater than

2000000_______

2 with 99  0's

Square of  3 with 99 0's will be

9000000_______

9 with 198  0's

Hence 199  digits

Square of  2 with 99 0's will be

4000000_______

4 with 198  0's

Hence 199  digits

Hence square of that number

is greater than a number of 199 digits also lesser than  a number of 199 digits

Hence number must be of 199 digits

so square of that number will have 199 Digits

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