Missing digits and divisibility: Class 8 practice
Divisibility tests find missing digits: by 3 or 9 the digit sum must be a multiple of 3 or 9; by 11 the difference between alternate-place digit sums must be a multiple of 11; by 4 the last two digits must form a multiple of 4.. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Divisibility rules
The idea behind missing digits and divisibility
Divisibility tests find missing digits: by 3 or 9 the digit sum must be a multiple of 3 or 9; by 11 the difference between alternate-place digit sums must be a multiple of 11; by 4 the last two digits must form a multiple of 4.
Step-by-step method
- Write the condition for the divisor.
- Express it with the unknown digit d (0–9).
- Find every d that works.
Worked example 1 (Basic)
Find all digits d that make 11d56 divisible by 9.
- Test the rule for 9 with d = 0 … 9
- Works for d = 5
- Answer: 5
Worked example 2 (Standard)
Find all digits d that make 1d1450 divisible by 11.
- Test the rule for 11 with d = 0 … 9
- Works for d = 3
- Answer: 3
Worked example 3 (Challenge)
Find all digits d that make 62415d2 divisible by 12.
- Test the rule for 12 with d = 0 … 9
- Works for d = 1, 7
- Answer: 1, 7
Common mistakes and misconceptions
- Forgetting d can be 0
- Missing a second valid digit
Practice questions
- (Basic) Find all digits d that make 44d97 divisible by 9.
- (Basic) Find all digits d that make 3d062 divisible by 3.
- (Standard) Find all digits d that make 181d46 divisible by 11.
- (Standard) Find all digits d that make 59985d divisible by 11.
- (Challenge) Find all digits d that make 55320d6 divisible by 12.
- (Challenge) Find all digits d that make 948d452 divisible by 4.
Answer key with full working
- Q1: Test the rule for 9 with d = 0 … 9 → Works for d = 3 → Answer: 3
- Q2: Test the rule for 3 with d = 0 … 9 → Works for d = 1, 4, 7 → Answer: 1, 4, 7
- Q3: Test the rule for 11 with d = 0 … 9 → Works for d = 3 → Answer: 3
- Q4: Test the rule for 11 with d = 0 … 9 → Works for d = 2 → Answer: 2
- Q5: Test the rule for 12 with d = 0 … 9 → Works for d = 3, 9 → Answer: 3, 9
- Q6: Test the rule for 4 with d = 0 … 9 → Works for d = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 → Answer: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Syllabus note
This skill is listed under "Playing with Numbers" for Class 8 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is missing digits and divisibility in?
"Playing with Numbers", Class 8 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.