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

Hint:

For “smallest number to subtract to be divisible by $m$”, compute remainder $r$ of $N$ on division by $m$ and subtract $r$.

Answer 1

The Correct Option is B

Find the remainder when $2085$ is divided by $23$.
$23\times 90=2070$, so remainder $=2085-2070=15$.
Subtracting the remainder makes it divisible by $23$: $2085-15=2070$.
Hence the least number to subtract is 15.

Answer 2

We need the smallest number that must be subtracted from 2085 to make it exactly divisible by 23. We can check each option by subtracting it from 2085 and testing whether the result divides evenly by 23.

1. 9: \( 2085-9=2076 \). Dividing by 23 gives \( 2076 \div 23=90.26 \), not a whole number, so 9 does not work.
2. 15: \( 2085-15=2070 \). Dividing by 23 gives \( 2070 \div 23=90 \) exactly, since \( 23 \times 90=2070 \), so 15 works.
3. 20: \( 2085-20=2065 \). Dividing by 23 gives \( 2065 \div 23=89.78 \), not a whole number, so 20 does not work.
4. 19: \( 2085-19=2066 \). Dividing by 23 gives \( 2066 \div 23=89.83 \), not a whole number, so 19 does not work.

Only subtracting 15 from 2085 gives 2070, which divides evenly by 23, and it is the smallest such number.

Therefore, the correct answer is 15.

Answer 3

We can find the remainder of 2085 on division by 23 by splitting the number into more manageable parts, 2000 and 85, finding the remainder of each separately, and combining them. We can then check each option against this combined remainder.

1. 9: Since \( 23 \times 86=1978 \), 2000 leaves a remainder of \( 2000-1978=22 \) on division by 23; and since \( 23 \times 3=69 \), 85 leaves a remainder of \( 85-69=16 \). Combining, \( 22+16=38 \), and \( 38-23=15 \), so the true remainder is 15, not 9.
2. 15: Following the same split, the combined remainder works out to \( (22+16)-23=15 \), matching this option exactly.
3. 20: The combined remainder from the split is 15, not 20.
4. 19: The combined remainder from the split is 15, not 19.

Splitting 2085 into 2000 and 85 and combining their individual remainders on division by 23 gives an overall remainder of 15, which is the smallest number that must be subtracted.

Therefore, the correct answer is 15.