Question

The smallest number that should be subtracted from 2085, so that the new number is completely divisible by 23 is

• A. 9
• B. 15
• C. 20
• D. 19

Answer 1

The Correct Option is B

The correct option is (B): 20.

Answer 2

We need the smallest number to subtract from 2085 so the result divides evenly by 23. We can check this directly by subtracting each option from 2085 and testing divisibility by 23.

1. Option (A): 9: 2085 - 9 = 2076. Dividing 2076 by 23 gives 90.26, which is not a whole number, so 2076 is not divisible by 23.
2. Option (B): 15: 2085 - 15 = 2070. Dividing 2070 by 23 gives exactly 90, a whole number, so 2070 is divisible by 23.
3. Option (C): 20: 2085 - 20 = 2065. Dividing 2065 by 23 gives 89.78, not a whole number, so this does not divide evenly.
4. Option (D): 19: 2085 - 19 = 2066. Dividing 2066 by 23 gives 89.8, again not a whole number, so this fails too.

Only subtracting 15 leaves 2070, which is exactly divisible by 23, and it is the smallest such number since it is the remainder left when 2085 is divided by 23.

Therefore, the correct answer is 15.

Answer 3

To find the smallest number that must be subtracted from 2085 to make it exactly divisible by 23, divide 2085 by 23 directly and look at the remainder, since subtracting that remainder is what brings the number down to the nearest multiple of 23 below it. Dividing, \( 23 \times 90 = 2070 \), and \( 2085 - 2070 = 15 \), so 2085 leaves a remainder of exactly 15 when divided by 23. Subtracting this remainder is the smallest possible subtraction that restores exact divisibility, since any smaller subtraction still leaves a non-zero remainder and any larger one overshoots past the nearest multiple of 23. Now compare each option against this remainder of 15.

1. Option (A): 9: Since the true remainder on dividing 2085 by 23 is 15, subtracting only 9 leaves 2076, and \( 2076 = 23 \times 90 + 6 \), a remainder of 6, so this is not enough to reach a multiple of 23.
2. Option (B): 15: This equals the remainder found by dividing 2085 by 23, so subtracting it lands exactly on \( 23 \times 90 = 2070 \), a multiple of 23, with nothing left over.
3. Option (C): 20: This is more than the required remainder of 15, so subtracting 20 gives 2065, and \( 2065 = 23 \times 89 + 18 \), still a remainder of 18, so it does not divide evenly, and even if it did, it would not be the smallest subtraction since 15 already works.
4. Option (D): 19: Also larger than the necessary remainder of 15, subtracting 19 gives 2066, and \( 2066 = 23 \times 89 + 19 \), a remainder of 19, so this too leaves a remainder and is unnecessarily large besides.

Dividing 2085 by 23 directly shows the remainder is exactly 15, and subtracting that remainder is both necessary and sufficient, and the smallest such subtraction.

Therefore, the correct answer is 15.