Division with Remainders • Topic 3 of 3

Divisibility and Finding Quotient Patterns

A number is exactly divisible by another when the remainder is 0. Knowing divisibility rules helps you divide quickly.

Quick Divisibility Rules:

  • Divisible by 2: last digit is 0,2,4,6,8
  • Divisible by 5: last digit is 0 or 5
  • Divisible by 9: sum of digits is divisible by 9
  • Divisible by 10: last digit is 0

The relationship between division and multiplication: if A / B = C, then B x C = A.

Divisibility Quick RulesBy 2Last digit:0,2,4,6,8By 5Last digit:0 or 5By 9Sum of digitsdivisible by 9By 10Last digit: 0If A / B = C then B x C = A (inverse relationship)Example: 648 / 9 = 72 because 9 x 72 = 648
Solved Examples
1
Worked Example
Is 648 divisible by 9? Verify by division.
SolutionSum of digits: 6+4+8=18. 18 is divisible by 9. So 648 is divisible by 9. Verify: 648 / 9 = 72 with remainder 0. Check: 9 x 72 = 648.
2
Worked Example
A number when divided by 7 gives quotient 58 and remainder 4. Find the number.
SolutionUsing the formula: Number = (Divisor x Quotient) + Remainder = (7 x 58) + 4 = 406 + 4 = 410.
3
Worked Example
Find the smallest 3-digit number exactly divisible by 8.
Solution100 / 8 = 12 R4. So 100 is not divisible by 8. To find the next: 8 x 13 = 104. Check: 104 / 8 = 13 R0. Answer: 104.

Key Points

  • A number with remainder 0 is exactly divisible by the divisor
  • Divisibility by 2: even last digit; by 5: last digit 0 or 5; by 9: digit sum divisible by 9
  • Formula: Number = (Divisor x Quotient) + Remainder
  • Division and multiplication are inverse operations
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.A number is divisible by $2$ if it:
Explanation: Even numbers.
Q2.A number is divisible by $5$ if it ends in:
Explanation: $0$ or $5$.
Q3.A number is divisible by $10$ if it ends in:
Explanation: Ends in $0$.
Q4."Divisible" means it divides exactly with:
Explanation: No remainder.
Practice more MCQs → 📝 Assignment · 35 marks