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About Playing with Numbers — Class 6 Olympiad

Master divisibility rules, prime factorization, HCF, LCM, and solve tricky Olympiad problems using number properties. This topic is part of the Olympiad Class 6 mathematics syllabus (chapter: Module 3). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Playing with Numbers — solved examples for Class 6 Olympiad

Example 1easy

What is the smallest 4-digit number that can be formed using four distinct non-zero digits, such that the digit in the tens place is twice the digit in the units place?
  1. A)1263
  2. B)1342
  3. C)2163
  4. D)1284

Step-by-step solution

  1. Identify possible pairs for (tens digit, units digit) where the tens digit is twice the units digit, using distinct non-zero digits: (2,1), (4,2), (6,3), (8,4).
  2. For each pair, construct the smallest 4-digit number by choosing the smallest available distinct non-zero digits for the thousands and hundreds places:
  3. - For (2,1): The remaining digits are from {3,4,5,6,7,8,9}. Smallest number is 3421.
  4. - For (4,2): The remaining digits are from {1,3,5,6,7,8,9}. Smallest number is 1342.
  5. - For (6,3): The remaining digits are from {1,2,4,5,7,8,9}. Smallest number is 1263.
  6. - For (8,4): The remaining digits are from {1,2,3,5,6,7,9}. Smallest number is 1284.
  7. Comparing 3421, 1342, 1263, and 1284, the smallest number is 1263.

Answer: 1263

Example 2medium

A six-digit number 5P6Q2R is divisible by 3, 4, and 5. If P and Q are distinct prime digits, what is the maximum possible value of P+Q+R?
  1. A)8
  2. B)10
  3. C)12
  4. D)15

Step-by-step solution

  1. Step 1: Apply divisibility rules for 5 and 4. For a number to be divisible by 5, its last digit R must be 0 or 5. For a number to be divisible by 4, its last two digits (2R) must be divisible by 4. If R=5, then 25 is not divisible by 4. If R=0, then 20 is divisible by 4. Therefore, R=0.
  2. Step 2: Apply divisibility rule for 3. The sum of the digits (5+P+6+Q+2+0) must be divisible by 3. This simplifies to 13+P+Q must be divisible by 3.
  3. Step 3: Identify distinct prime digits P and Q. Prime digits are 2, 3, 5, 7. We need to find distinct P, Q from this set such that 13+P+Q is a multiple of 3, and P+Q is maximized.
  4. Step 4: Test pairs for (P,Q) to maximize P+Q: If P+Q = 5 (e.g., P=2, Q=3), 13+5 = 18 (divisible by 3). P+Q+R = 5+0 = 5. If P+Q = 8 (e.g., P=3, Q=5), 13+8 = 21 (divisible by 3). P+Q+R = 8+0 = 8. No other distinct prime digit pairs yield a higher P+Q that makes 13+P+Q divisible by 3. The maximum valid sum P+Q is 8. Thus, the maximum possible value of P+Q+R is 8.

Answer: 8

Example 3hard

A 5-digit number 7P5Q2 is divisible by both 8 and 9. What is the smallest possible value of P + Q?
  1. A)2
  2. B)3
  3. C)4
  4. D)5

Step-by-step solution

  1. For 7P5Q2 to be divisible by 8, the number formed by its last three digits, 5Q2, must be divisible by 8. Possible values for Q are 1 (512 = 8 × 64), 3 (532 is not div by 8), 5 (552 = 8 × 69), 7 (572 is not div by 8), 9 (592 = 8 × 74). So, Q can be 1, 5, 9.
  2. For 7P5Q2 to be divisible by 9, the sum of its digits (7 + P + 5 + Q + 2 = 14 + P + Q) must be divisible by 9.
  3. We need the smallest possible value of P + Q. Let's test the possible values for Q:
  4. Case 1: If Q = 1, then 14 + P + 1 = 15 + P must be divisible by 9. The smallest P (0-9) for this is P = 3 (15 + 3 = 18). So, P + Q = 3 + 1 = 4.
  5. Case 2: If Q = 5, then 14 + P + 5 = 19 + P must be divisible by 9. The smallest P for this is P = 8 (19 + 8 = 27). So, P + Q = 8 + 5 = 13.
  6. Case 3: If Q = 9, then 14 + P + 9 = 23 + P must be divisible by 9. The smallest P for this is P = 4 (23 + 4 = 27). So, P + Q = 4 + 9 = 13.
  7. Comparing the P + Q values (4, 13, 13), the smallest possible value is 4.

Answer: 4

Practice questions on Playing with Numbers

  1. Q1.easy

    What is the smallest digit that can replace the asterisk (*) in the number 7*542 to make it divisible by 9?
    1. A)3
    2. B)6
    3. C)0
    4. D)9
    Show answer

    Answer: 0

    Hint: Remember the divisibility rule for 9: the sum of all digits must be divisible by 9.

  2. Q2.easy

    A 3-digit number has its units digit as 0. If this number is divisible by both 4 and 9, what is the sum of its hundreds and tens digits?
    1. A)7
    2. B)9
    3. C)12
    4. D)18
    Show answer

    Answer: 9

    Hint: Use the divisibility rules for 4 and 9 to narrow down the possibilities for the tens digit and then the hundreds digit.

  3. Q3.easy

    Which of the following statements about prime numbers is TRUE?
    1. A)All prime numbers are odd.
    2. B)The sum of any two prime numbers is always even.
    3. C)There is only one even prime number.
    4. D)Every odd number is a prime number.
    Show answer

    Answer: There is only one even prime number.

    Hint: Consider each statement carefully and try to find a counterexample if you think it's false. Remember the definition of prime numbers.

  4. Q4.medium

    Find the smallest natural number N such that N has exactly 12 factors and its prime factorization contains only the prime numbers 2 and 3.
    1. A)72
    2. B)96
    3. C)108
    4. D)144
    Show answer

    Answer: 72

    Hint: If N = p^a × q^b, the number of factors is (a+1)(b+1). To make N smallest, assign the larger exponent to the smaller prime base.

  5. Q5.medium

    A rectangular floor measures 1.2 meters by 1.68 meters. What is the minimum number of identical square tiles required to completely cover this floor without cutting any tiles?
    1. A)35
    2. B)42
    3. C)48
    4. D)60
    Show answer

    Answer: 35

    Hint: First, convert all measurements to centimeters. Then, the side length of the largest possible square tile will be the HCF of the floor's dimensions. Finally, calculate the number of tiles.

  6. Q6.medium

    Three different traffic lights change after every 48 seconds, 72 seconds, and 108 seconds respectively. If they all change simultaneously at 7:00:00 AM, how many additional times will they change simultaneously again in the next 30 minutes?
    1. A)3
    2. B)4
    3. C)5
    4. D)6
    Show answer

    Answer: 4

    Hint: Find the LCM of the intervals to determine when they will next change simultaneously. Then calculate how many such intervals fit into 30 minutes, excluding the initial change.

  7. Q7.hard

    A composite number N has exactly three factors. Which of the following statements about N must be true?
    1. A)N is a prime number.
    2. B)N is the square of a prime number.
    3. C)N is the product of two distinct prime numbers.
    4. D)N is the cube of a prime number.
    Show answer

    Answer: N is the square of a prime number.

    Hint: Recall the formula for the number of factors of a number based on its prime factorization. Consider what type of exponent structure would lead to exactly three factors.

  8. Q8.hard

    A group of students wants to arrange 120 pencils, 168 erasers, and 216 sharpeners into identical packs such that no item is left over. What is the maximum number of packs they can make?
    1. A)12
    2. B)18
    3. C)24
    4. D)36
    Show answer

    Answer: 24

    Hint: To make identical packs with no items left over, the number of packs must be a common factor of all three quantities. You need to find the largest such common factor.

  9. Q9.hard

    Three friends, Alex, Ben, and Chris, visit a library every 4 days, 6 days, and 9 days respectively. If they all met at the library on a Sunday, on which day of the week will they next meet together?
    1. A)Monday
    2. B)Tuesday
    3. C)Wednesday
    4. D)Thursday
    Show answer

    Answer: Monday

    Hint: First, find the Least Common Multiple (LCM) of their visiting intervals to determine when they will meet again. Then, use this number of days to calculate the day of the week.

These are 9 of the 60 questions available for Playing with Numbers. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.