Remainders & Modular Reasoning • Topic 1 of 4

Quotient & Remainder

When you divide an integer N by a positive divisor d, you get a quotient q and a remainder r that obey N = d·q + r, where 0 ≤ r < d. The remainder is never negative and never as large as the divisor — dividing by 5 can only leave 0, 1, 2, 3 or 4. The GRE calculator gives a decimal, not a remainder, so you convert: 47 ÷ 5 = 9.4 means the whole-number quotient is 9, and the remainder is 47 − 5×9 = 2. Reading it the other way is just as useful: "a number that leaves remainder 3 when divided by 7" is any number of the form 7k + 3 (3, 10, 17, 24, …). Being fluent in both directions — from a number to its remainder, and from a remainder to the family of numbers — is what these problems test.

✅ Solved examples

1. What is the remainder when 47 is divided by 6?
6 × 7 = 42, and 47 − 42 = 5. Since 5 < 6, the remainder is 5. (Check: 47 = 6·7 + 5.)
2. A number leaves remainder 3 when divided by 8. Write its general form and list two examples.
General form: 8k + 3. For k = 0, 1, 2 you get 3, 11, 19. Any of these works.
3. When 100 is divided by a number, the quotient is 7 and the remainder is 2. What is the divisor?
N = d·q + r → 100 = d·7 + 2 → 7d = 98 → d = 14.
4. Quantitative Comparison. n is a positive integer. Quantity A: the remainder when n is divided by 5. Quantity B: 5. Which is greater?
A remainder on division by 5 is at most 4 (it must be less than the divisor). So Quantity A ≤ 4 < 5 = Quantity B: Quantity B is greater.

✏️ Practice — try these, take hints as needed

1. Remainder when 83 is divided by 9?
9 × 9 = 81.
83 − 81.
Less than 9.
2
2. A number leaves remainder 4 on division by 7. Give its general form.
divisor × k + remainder.
7k + 4.
e.g. 4, 11, 18.
7k + 4
3. Dividing N by 12 gives quotient 5 and remainder 7. Find N.
N = 12·5 + 7.
60 + 7.
Add.
67
4. What is the largest possible remainder when dividing by 15?
Remainder < divisor.
Max is divisor − 1.
15 − 1.
14
5. Remainder when 250 is divided by 8?
8 × 31 = 248.
250 − 248.
Less than 8.
2

📝 Topic test — 8 questions

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