Lesson 6 of 100 | Quantitative Aptitude / Number System / संख्या पद्धति
Remainders in Elementary Number Problems
Learning outcome
Express division with a valid remainder and simplify elementary remainder problems involving sums, differences and products.
Concepts and reasons
Here the dividend N is a nonnegative (zero or positive) integer and divisor d is a positive integer. Write N = d × q + r: q is the whole-number quotient and r is the remainder, with 0 ≤ r < d.
The remainder is what remains after removing the maximum number of complete groups of size d. It cannot equal or exceed d: that would allow another complete group. A remainder of 0 means exact divisibility. When N < d, the quotient is 0 and remainder is N.
For the same divisor, replace numbers by their remainders when finding the remainder of a sum, difference or product. Then reduce the result again into the allowed range.
Why does replacement work? Write A = d × a + r and B = d × b + s. Adding or subtracting leaves r + s or r - s apart from multiples of d. Multiplication leaves r × s apart from multiples of d. Those complete multiples contribute no remainder.
A negative intermediate difference is not the final remainder under our convention. Add d to put it into the allowed range.
Worked examples
Example 1. Divide 157 by 12. Since 12 × 13 = 156, write 157 = 12 × 13 + 1. The remainder is 1, which lies between 0 and 11.
Example 2. A and B leave remainders 5 and 4 when divided by 7. Their sum leaves the remainder of 5 + 4 = 9, namely 2. Their product leaves the remainder of 5 × 4 = 20, namely 6.
Example 3. Suppose A ≥ B and their remainders upon division by 7 are 2 and 5. For A - B, the provisional difference is -3. Add 7: the required remainder is 4, not -3.
Common traps
A remainder is not a quotient or a decimal part. Divide oversized results again to find the remainder. Do not combine remainders obtained using different divisors, and never divide by zero.
Practice questions
- Find the remainder when 246 is divided by 11.
- N leaves remainder 5 upon division by 8. Find the remainder of 3N + 7 upon division by 8.
- N leaves remainder 3 upon division by 5. Find the remainder of (N + 1)(N + 2) upon division by 5.
Answers and explanations
- Since 11 × 22 = 242, write 246 = 11 × 22 + 4. The remainder is 4, smaller than 11.
- Replace N by 5: 3 × 5 + 7 = 22. Since 22 = 8 × 2 + 6, the remainder is 6.
- N + 1 leaves 4, while N + 2 leaves 0. Their product therefore leaves 4 × 0 = 0.