What is the square root of 18769 using digital sum formula options are-a(133) b(137) c(143) d(147)

The digital sum (or digital root) is found by repeatedly summing the digits until a single digit is obtained.
$DS(18769) = 1 + 8 + 7 + 6 + 9 = 31$
$DS(31) = 3 + 1 = 4$
So, the digital sum of 18769 is 4. If a number is a perfect square, its digital sum must be 1, 4, 7, or 9. Our number's digital sum is 4, which is consistent.

If $x$ is the square root of $N$, then $DS(N) = DS((DS(x))^2)$. We will check this for each option.

• a) 133 $DS(133) = 1 + 3 + 3 = 7$ $DS((DS(133))^2) = DS(7^2) = DS(49) = 4 + 9 = 13$ $DS(13) = 1 + 3 = 4$. This matches the digital sum of 18769.

• b) 137 $DS(137) = 1 + 3 + 7 = 11$ $DS(11) = 1 + 1 = 2$ $DS((DS(137))^2) = DS(2^2) = DS(4) = 4$. This matches the digital sum of 18769.

• c) 143 $DS(143) = 1 + 4 + 3 = 8$ $DS((DS(143))^2) = DS(8^2) = DS(64) = 6 + 4 = 10$ $DS(10) = 1 + 0 = 1$. This does not match the digital sum of 18769 (which is 4).

• d) 147 $DS(147) = 1 + 4 + 7 = 12$ $DS(12) = 1 + 2 = 3$ $DS((DS(147))^2) = DS(3^2) = DS(9) = 9$. This does not match the digital sum of 18769 (which is 4).

Options c) and d) are eliminated because their squared digital sums do not match the digital sum of 18769. Options a) and b) remain as possibilities.

The number 18769 ends in 9. For a perfect square ending in 9, its square root must end in either 3 or 7.

• Option a) 133 ends in 3.
• Option b) 137 ends in 7. Both remaining options satisfy this condition.

Since both the digital sum formula and the last digit property leave two possible answers, we perform a direct calculation for the remaining options to find the correct square root.

• Check option a) 133: $133^2 = 133 \times 133 = 17689$ (This is not 18769)
• Check option b) 137: $137^2 = 137 \times 137 = 18769$ (This matches the given number)

Therefore, the square root of 18769 is 137.

The final answer is $\boxed{\text{137}}$.