Q1. The sum of four consecutive numbers is 34. The numbers are:

1. 6, 7, 8, 9
2. 7, 8, 9, 10
3. 5, 8, 9, 12
4. 4, 9, 10, 11

Q2. Which expression is always even for every integer value of n?

1. 3n + 5
2. 4n + 2
3. 5n + 1
4. n² + 1

Q3. Which number leaves remainder 3 when divided by 5?

1. 5k + 2
2. 5k + 3
3. 3k + 5
4. 5k − 3

Q4. If a number is divisible by both 9 and 4, then it must be divisible by:

1. 12
2. 18
3. 36
4. 45

Q5. Which number is divisible by 9?

1. 4532
2. 7209
3. 8312
4. 6254

Q6. A number is divisible by 11 when the difference between the sums of alternate digits is:

1. 1
2. 5
3. 11 or a multiple of 11
4. 9

Q7. The digital root of 5832 is:

1. 6
2. 8
3. 9
4. 18

Q8. In a cryptarithm, the same letter represents:

1. different digits in different places
2. the same digit throughout
3. only even digits
4. only prime digits

Q9. In a cryptarithm, the first digit of a number:

1. may always be 0
2. must be 0
3. can never be 0
4. must be 9

Q10. If a divides M and a divides N, then a also divides:

1. only M + N
2. only M − N
3. both M + N and M − N
4. neither

Q11. Write the next three numbers in the sequence: 17, 22, 27, 32, ______, ______, ______

Q12. Four consecutive integers have the form: n, n + 1, n + 2, n + 3. If their sum is 54, find the four numbers.

Q13. Without actually calculating, determine whether each expression is even or odd:

• 8m + 6
• 7p + 4
• 12q − 8
• 5r + 7

Q14. Find the remainder when:

• 47 is divided by 5
• 83 is divided by 7
• 125 is divided by 6

Q15. Write the general form of all numbers that leave remainder 2 when divided by 5.

Q16. Using the divisibility rule for 9, determine whether 72,459 is divisible by 9. Show your reasoning.

Q17. Find the digital root of:

• 48,729
• 99,999
• 12,345

Q18. Give one example and one explanation for the statement: “Every multiple of 6 is a multiple of 3.”

Q19. Consecutive Numbers: The sum of five consecutive numbers is 125. Find all five numbers and verify your answer.

Q20. Parity Reasoning: Four consecutive integers are represented by n, n + 1, n + 2, n + 3. Prove that the sum of these four numbers is always even.

Q21. Always, Sometimes or Never: Determine whether each statement is Always True, Sometimes True or Never True. Give an example or counterexample.

1. The sum of two even numbers is divisible by 4.
2. The sum of two multiples of 4 is a multiple of 4.
3. If a number is divisible by 7, it is divisible by every multiple of 7.

Q22. Remainders: A number leaves remainder 3 when divided by 7. Another number leaves remainder 5 when divided by 7. Without finding the actual numbers, determine the remainder when their:

1. sum is divided by 7
2. difference is divided by 7

Explain your reasoning.

Q23. Divisibility: Find all possible values of x so that 52×4 is divisible by 9. Show the reasoning using the divisibility rule.

Q24. Divisibility by 11: Determine whether each number is divisible by 11. If it is not, find the remainder using the divisibility shortcut.

• 841
• 5529
• 90,904

Q25. A number leaves remainder 2 when divided by 3, remainder 3 when divided by 4, and remainder 4 when divided by 5. Find the smallest positive number satisfying all three conditions. Explain why your answer is the smallest.

Q26. Consider the statement: “If two numbers are not divisible by 6, their sum cannot be divisible by 6.” Is this statement Always True, Sometimes True or Never True? Give examples and an explanation.

Q27. Solve the cryptarithm:

Find the values of A and B.

Q28. Solve:

Find A and B.

Q29. Solve:

Find P, Q, R and S.

Q30. Solve:

Find A, B and C.

Q31. Challenge: Solve:

Find J and K.

⭐ Bonus Challenge – Q32. Solve the cryptarithm:

Find all possible values of E, F and G.