CAT 2025 Slot 3 · Quant · Question 17

Deal with the subtraction first, since only {10}^{25} is affected: the expression is {10}^{50}+\left({10}^{25}-123 \right).

Now {10}^{25}-1 is a string of 25 nines, so {10}^{25}-123 is that minus 122.

Subtracting 122 from 25 nines changes only the last three digits, giving 22 nines followed by 877.

That is a 25-digit number, while {10}^{50} is a 1 followed by 50 zeros.

Adding them places the 25-digit block in the last 25 positions, so the number is 1, then 25 zeros, then those 25 digits.

The digit sum is 1+0+\left( 22\times 9 \right)+8+7+7.

That is 1+198+22=221.

Shortcut. Never write the number out. Subtracting a small number from a power of 10 always produces a run of nines followed by a short tail, and adding a much larger power of 10 simply puts a 1 in front with zeros between — so the digit sum is one 1, a block of nines, and the tail.