Explore what stays true when signs change, numbers are grouped, or digits are added. This set of MCQs with answers and explanations follows Class 8 Maths, Ganita Prakash Part 1, Chapter 5: Number Play. Practise parity, divisibility, remainders, digital roots and cryptarithms through numerical examples and algebraic reasoning. The explanations show how to justify a pattern, apply a divisibility test and track carries in a digit puzzle.
Class 8 Maths Ganita Prakash Part 1 Chapter 5 MCQs with Answers
Set 1: Consecutive Integers, Signs and Parity MCQs (Questions 1–5)
Q1. Four consecutive positive integers have a sum of 34. What is the smallest of these integers?
[A]. 8
[B]. 6
[C]. 9
[D]. 7
Q2. In the expression 3 □ 4 □ 5 □ 6, each box can independently contain either + or −. How many different arrangements of signs are possible?
[A]. 6
[B]. 3
[C]. 16
[D]. 8
Q3. Four consecutive integers are written in order, with either + or − inserted between each pair. What can be said about the parity of every resulting value?
[A]. It depends on which integer comes first.
[B]. It is odd only when all three signs are +.
[C]. It is always even.
[D]. It is always odd.
Q4. In an expression, a term +b is changed to −b while every other term stays the same. What is the new value minus the original value?
[A]. 2b
[B]. −2b
[C]. 0
[D]. −b
Q5. Which expression is even for every pair of integers m and n?
[A]. m² + 1
[B]. m² + 2
[C]. 4m + 2n
[D]. 3m + 5n
Set 2: Divisibility and Common Remainders MCQs (Questions 6–10)
Q6. Two even integers each leave a remainder of 2 when divided by 4. What must be true about their sum?
[A]. It leaves a remainder of 2 when divided by 4.
[B]. It leaves a remainder of 1 when divided by 4.
[C]. It is divisible by 4.
[D]. It is odd.
Q7. M and N are positive integers that are both divisible by 8. Which statement must be true?
[A]. M − N is never divisible by 8.
[B]. M ÷ N is always divisible by 8.
[C]. Both M + N and M − N are divisible by 8.
[D]. M + N is odd.
Q8. A positive integer is divisible by both 9 and 4. Which number must also divide it?
[A]. 72
[B]. 81
[C]. 36
[D]. 144
Q9. A positive integer leaves a remainder of 2 when divided by 3 and also when divided by 4. Which expression describes all such integers, with k a non-negative integer?
[A]. 12k + 2
[B]. 12k + 1
[C]. 12k + 6
[D]. 12k + 4
Q10. A bag contains fewer than 100 pebbles. Grouping them in twos, threes or fives leaves one pebble each time. Grouping them in sevens leaves none. How many pebbles are there?
[A]. 31
[B]. 91
[C]. 61
[D]. 71
Set 3: Remainders and Multiples of 9 MCQs (Questions 11–15)
Q11. Each of three positive integers leaves a remainder of 2 when divided by 6. What is the remainder when their sum is divided by 6?
[A]. 3
[B]. 0
[C]. 1
[D]. 2
Q12. When divided by 7, 661 leaves a remainder of 3 and 4779 leaves a remainder of 5. What remainder does 661 + 4779 leave when divided by 7?
[A]. 2
[B]. 5
[C]. 8
[D]. 1
Q13. What is the smallest positive integer that leaves remainders 2, 3 and 4 when divided by 3, 4 and 5 respectively?
[A]. 29
[B]. 60
[C]. 59
[D]. 119
Q14. Which of the following numbers is divisible by 9?
[A]. 8888
[B]. 93547
[C]. 405
[D]. 123
Set 4: Divisibility Tests and Multiple Patterns MCQs (Questions 16–20)
Q16. How many multiples of 9 lie strictly between 4300 and 4400?
[A]. 11
[B]. 9
[C]. 10
[D]. 12
Q18. What is the smallest positive multiple of 9 whose digits are all even?
[A]. 198
[B]. 288
[C]. 180
[D]. 270
Q20. Which pair of divisibility tests is sufficient to show that an integer is divisible by 24?
[A]. Divisibility by both 3 and 8
[B]. Divisibility by both 2 and 12
[C]. Divisibility by both 4 and 6
[D]. Divisibility by both 3 and 4
Set 5: Digital Roots and Cryptarithms MCQs (Questions 21–25)
Q21. The digital root is the single digit obtained by repeatedly adding a positive integer’s digits. What is the digital root of 489710?
[A]. 8
[B]. 1
[C]. 9
[D]. 2
Q22. What is the digital root of every positive multiple of 9?
[A]. It depends on the number of digits.
[B]. 0
[C]. 3
[D]. 9
Q23. An eight-digit positive integer has digital root 5. What is the digital root of the number obtained by adding 10 to it?
[A]. 5
[B]. 6
[C]. 7
[D]. 1
Q24. For non-negative integers a and b, what is the digital root of 9a + 36b + 13?
[A]. 4
[B]. 9
[C]. 1
[D]. 7
Q25. In the cryptarithm A1 + 1B = B0, each letter stands for a different digit and leading digits are non-zero. What are A and B?
[A]. A = 7, B = 9
[B]. A = 9, B = 7
[C]. A = 7, B = 8
[D]. A = 8, B = 9
Set 6: Digit Puzzles and Divisibility Patterns MCQs (Questions 26–30)
Q26. In the cryptarithm AB + 37 = 6A, different letters stand for different digits and A is non-zero. What is the two-digit number AB?
[A]. 35
[B]. 52
[C]. 23
[D]. 25
Q27. The four-digit number 31z5 is divisible by 9, where z is a digit. Which list gives all possible values of z?
[A]. 3 and 6
[B]. 0 and 9
[C]. 9 only
[D]. 0 only
Q28. For which pair of digits (a, b) is the six-digit number 48a23b divisible by 18?
[A]. (4, 4)
[B]. (5, 4)
[C]. (6, 4)
[D]. (6, 5)
Q29. Which statement about the sum of three consecutive even integers is always true?
[A]. It is divisible by 6.
[B]. It is odd.
[C]. It is divisible by 12.
[D]. It is divisible by 9.
Q30. Which statement correctly describes the relationship among multiples of 4, 8 and 32?
[A]. Every multiple of 32 is a multiple of 8, and every multiple of 8 is a multiple of 4.
[B]. Every multiple of 8 is a multiple of 32.
[C]. No multiple of 32 is a multiple of 4.
[D]. Every multiple of 4 is a multiple of 32.
Part 1 Chapter 5: Full Chapter MCQs (Set 1)
Q31. Which numbers can be written as a sum of two consecutive numbers?
[A]. All even numbers
[B]. All odd numbers
[C]. All multiples of 4
[D]. Only numbers greater than 10
Q32. When placing ‘+’ and ‘-‘ signs between 4 consecutive numbers, how many different possibilities for expressions exist?
[A]. 4
[B]. 6
[C]. 8
[D]. 16
Q33. When 4 consecutive numbers are chosen, and ‘+’ or ‘-‘ signs are placed between them, what is always true about the parity of the resulting expressions?
[A]. They are always odd.
[B]. They alternate between even and odd.
[C]. They always have even parity.
[D]. Their parity varies randomly.
Q34. If the difference between two numbers is an even number, what can be concluded about their parities?
[A]. Both must be odd.
[B]. Both must be even.
[C]. They must have different parities.
[D]. Either both are even or both are odd.
Part 1 Chapter 5: Full Chapter MCQs (Set 2)
Q35. What is the parity of odd ± odd?
[A]. Odd
[B]. Even
[C]. Cannot be determined
[D]. Depends on the numbers chosen
Q36. Which of the following algebraic expressions will always evaluate to an even number for any integer values of m and q?
[A]. 3g + 5h
[B]. x² + 2
[C]. 4m + 2q
[D]. b² + 1
Q37. When is the sum of two even numbers a multiple of 4?
[A]. Only if both numbers are multiples of 4.
[B]. Only if both numbers are not multiples of 4.
[C]. If both numbers are multiples of 4 or if both numbers are not multiples of 4.
[D]. Never
Q38. . If ‘a’ divides ‘M’ and ‘a’ divides ‘N’, what can be concluded about M + N and M – N?
[A]. They will never be multiples of ‘a’.
[B]. They will also be multiples of ‘a’.
[C]. Their divisibility depends on the values of M and N.
[D]. They will sometimes be multiples of ‘a’.
Part 1 Chapter 5: Full Chapter MCQs (Set 3)
Q39. If a number is divisible by 8, is it always true that 8 also divides any two numbers (separately) that add up to that number?
[A]. Always true
[B]. Sometimes true
[C]. Never true
[D]. Only if the two numbers are also multiples of 8
Q40. If ‘A’ is divisible by ‘k’, then all multiples of ‘A’ are divisible by ‘k’. Is this statement:
[A]. Always true
[B]. Sometimes true
[C]. Never true
[D]. Only if ‘k’ is a prime number
Q41. If a number is divisible by 12, what else is it always divisible by?
[A]. Only 2 and 6
[B]. Only 3 and 4
[C]. All the factors of 12
[D]. Only 12 itself
Q42. If a number is divisible by both ‘k’ and ‘m’, what else is it divisible by?
[A]. The product of k and m.
[B]. The sum of k and m.
[C]. The LCM of k and m.
[D]. The difference between k and m.
Part 1 Chapter 5: Full Chapter MCQs (Set 4)
Q43. When you add an odd number to an even number, you get a multiple of 6. Is this statement:
[A]. Always true
[B]. Sometimes true
[C]. Never true
[D]. Only if the odd number is 1
Q44. Which algebraic expression captures all numbers that leave a remainder of 3 when divided by 5?
[A]. 3k – 5
[B]. 5k + 3
[C]. 5k – 3
[D]. 3k + 5
Q45. What is the divisibility shortcut for a number being divisible by 10?
[A]. Its first digit must be 0.
[B]. Its last digit (units digit) must be 0.
[C]. The sum of its digits must be a multiple of 10.
[D]. It must be an even number.
Q46. How is the remainder found when a number is divided by 9, using the shortcut method?
[A]. By looking at the units digit.
[B]. By subtracting 9 from the number.
[C]. By adding the digits of a number repeatedly until a single digit is obtained.
[D]. By dividing the number by 3, then by 3 again.
Part 1 Chapter 5: Full Chapter MCQs (Set 5)
Q47. What is the shortcut for divisibility by 3?
[A]. The number must end in 0, 3, 6, or 9.
[B]. The sum of its digits must be divisible by 3.
[C]. It must be an even number.
[D]. The product of its digits must be divisible by 3.
Q48. What is the alternating pattern observed when powers of 10 are divided by 11?
[A]. All remainders are 0.
[B]. Remainders alternate between 1 more and 1 less than a multiple of 11.
[C]. Remainders are multiples of 10.
[D]. All remainders are 1.
Q49. How do you check divisibility by 6?
[A]. Check divisibility by 2 and 4.
[B]. Check divisibility by 2 and 3.
[C]. Check divisibility by 3 and 4.
[D]. Check divisibility by 1 and 6.
Q50. What is a “digital root” of a number?
[A]. The first digit of the number.
[B]. The sum of its digits, calculated once.
[C]. The single-digit number obtained by repeatedly adding its digits.
[D]. The number of digits in the number.
Explain Remainders and Check Digit Puzzles
For a remainder problem, write the number as divisor × quotient + remainder, with a non-negative remainder smaller than the divisor. Link digital roots to remainders on division by 9, remembering that positive multiples of 9 have digital root 9. In cryptarithms, track each carry and verify the completed addition with the chosen digits.
