Hard Level Worksheet
Very Short Answer Questions (1 Mark Each)
(1) Write the smallest 3-digit number divisible by both 3 and 5.
Correct! LCM of 3 and 5 = 15. Smallest 3-digit multiple of 15 is 105.
(2) Write the largest 4-digit number divisible by 7.
Perfect! 9999 ÷ 7 = 1428 remainder 3, so 9999 - 3 = 9996.
(3) Write all prime factors of 84.
(4) Write the HCF of 45, 60, and 75.
Excellent! HCF of 45, 60, and 75 is 15.
(5) Find the LCM of 16 and 20.
Great! 16 =
Short Answer Questions (2 Marks Each)
Note: Answer each question with complete working and clear explanations.
(1) Find the HCF of 72 and 108 using the prime factorisation method. HCF =
Perfect! HCF takes the lowest powers of common prime factors.
(2) Find the LCM of 24 and 36 by the division method. LCM =
(3) Check whether 7,524 is divisible by 6.
Yes, 7,524 is divisible by 6.
(4) Write all factors of 48 and find their sum. Sum:
Excellent! The sum of all factors of 48 is 124.
(5) A number is divisible by both 4 and 9. Write the smallest such number greater than 100.
Correct! 108 is the smallest multiple of 36 greater than 100.
Long Answer Questions (4 Marks Each)
Note: Answer each question with complete working and clear explanations.
(1) A farmer has 150 mangoes and 180 oranges. He wants to pack them into boxes so that each box has the same number of fruits and no fruit is left over. Find the greatest number of fruits that can be placed in each box.
Perfect! Greatest number of fruits per box = 30.
(2) Find the HCF and LCM of 45, 60, and 75 using the prime factorisation method, and verify the relation: HCF × LCM = Product of the numbers. HCF =
Note: The relation HCF × LCM = Product works only for two numbers, not three.
(3) Three bells ring at intervals of 15 minutes, 20 minutes, and 25 minutes. If they ring together at 8:00 a.m., at what time will they next ring together?
Next time together: 8:00 a.m. + 5 hours = 1:00 p.m.
(4) Find the smallest 4-digit number divisible by 12, 15, and 18. Smallest 4-digit number =
Excellent! The smallest 4-digit number divisible by 12, 15, and 18 is 1080.
(5) The traffic lights at three different road crossings change after every 48 seconds, 72 seconds, and 108 seconds respectively. If they change together at 8
00 a.m., find the time when they will change together again.
Next time together: 8 : 00 : 00 a.m. + 7 min 12 sec = 8 : 07 : 12 a.m.
Part B: Objective Questions (1 Mark Each)
Choose the correct answer and write the option (a/b/c/d)
(1) The smallest number divisible by both 8 and 12 is:
(a) 12 (b) 24 (c) 48 (d) 96
Correct! LCM of 8 and 12 = 24.
(2) The LCM of 36 and 48 is:
(a) 72 (b) 96 (c) 144 (d) 192
Correct! 36 =
(3) Which of the following numbers is divisible by 11?
(a) 2,431 (b) 2,442 (c) 2,453 (d) 2,462
Correct! Using alternating sum: (2+3)-(4+1) = 0, divisible by 11.
(4) The HCF of 64 and 96 is:
(a) 16 (b) 24 (c) 32 (d) 48
Correct! 64 =
(5) The smallest 3-digit number divisible by 8 is:
(a) 100 (b) 104 (c) 108 (d) 112
Correct! 104 = 8 × 13 is the smallest 3-digit multiple of 8.
(6) The LCM of 4, 6, and 9 is:
(a) 18 (b) 36 (c) 72 (d) 108
Correct! LCM =
(7) Which of the following numbers is a common multiple of 5 and 12?
(a) 60 (b) 65 (c) 72 (d) 84
Correct! 60 = 5 × 12 = LCM of 5 and 12.
(8) The HCF of 27, 45, and 63 is:
(a) 3 (b) 6 (c) 9 (d) 15
Correct! All three numbers are multiples of 9: 27 =
(9) Which is the smallest number that leaves a remainder of 3 when divided by each of 4, 5, and 6?
(a) 63 (b) 123 (c) 63 (d) 123
Correct! LCM(4,5,6) = 60, so number = 60 + 3 = 63.
(10) The LCM of two numbers is 180 and their HCF is 15. If one number is 45, the other number is:
(a) 45 (b) 60 (c) 75 (d) 90
Correct! Using HCF × LCM = Product: 15 × 180 = 45 × other number, so other = 60.
Complex HCF and LCM Properties
Determine whether these statements are True or False:
Playing With Numbers - Hard Quiz
🎉 Exceptional Mastery! Advanced Number Theory Conquered:
You have successfully mastered the "Playing With Numbers (Hard)" worksheet and achieved:
(1) Advanced HCF Calculations: Computing highest common factors for multiple numbers using prime factorization methods
(2) Complex LCM Computations: Finding least common multiples for three or more numbers systematically
(3) Multi-step Divisibility Analysis: Applying advanced divisibility rules including the rule for 11 and combined conditions
(4) Prime Factorization Mastery: Breaking down large numbers into prime factors and using them for HCF/LCM calculations
(5) Real-world Problem Applications: Solving complex scenarios involving time intervals, packaging problems, and synchronization
(6) Mathematical Relationship Understanding: Applying the fundamental relationship HCF × LCM = Product of two numbers
(7) Advanced Number Classification: Finding specific numbers meeting multiple divisibility criteria simultaneously
(8) Time and Interval Calculations: Converting between different time units and finding periodic repetitions
(9) Optimization Problems: Determining maximum/minimum values in practical distribution scenarios
(10) Systematic Problem-Solving: Using organized approaches for multi-constraint number problems
(11) Verification Techniques: Checking mathematical relationships and validating solutions through multiple methods
(12) Pattern Recognition: Identifying cyclical patterns in time-based problems and number sequences
(13) Complex Factorization: Working with numbers having multiple prime factors and finding their relationships
(14) Advanced Logical Reasoning: Combining multiple mathematical concepts to solve sophisticated word problems
(15) Strategic Mathematical Thinking: Choosing appropriate methods (listing, prime factorization, division) based on problem requirements
Outstanding achievement! You've mastered advanced number theory with exceptional mathematical sophistication!