Calculating Digits in Square Root

To determine the number of digits in the square root of an integer like 18769, use a straightforward digit-pairing method.

Digit Pairing Method for Square Roots

Method: Starting from the rightmost digit, group the digits of the number into pairs. The total count of these pairs (or the single digit if the leftmost group contains only one digit) indicates the number of digits in the square root.

Applying the Method to 18769

Consider the number 18769:

• The number has 5 digits.
• Group the digits in pairs from the right: $1$ $87$ $69$.
• There are 3 groups formed: one single digit ('1') and two pairs ('87', '69').

Based on this grouping, the square root of 18769 must have 3 digits.

Mathematical Verification

We can confirm this result:

• We know that $100^2 = 10000$ (a 4-digit number) and $200^2 = 40000$ (a 5-digit number).
• Since 18769 lies between 10000 and 40000 ($10000 < 18769 < 40000$), its square root must be between 100 and 200 ($100 < \sqrt{18769} < 200$).
• Any integer between 100 and 199 (inclusive) contains exactly 3 digits.
• Indeed, the exact square root is $\sqrt{18769} = 137$, which is a 3-digit number.

Therefore, the square root of 18769 has 3 digits.