Hard Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

This hard level explores advanced divisibility tests including division by 7, complex HCF/LCM problems, and multi-digit number theory.

Master these concepts for competitive examinations and advanced problem-solving.

1. Write the prime factorization of 315.

315 = 3² × 5 × 73 × 5 × 7 or

Perfect! 315 = 3² × 5 × 7 = 9 × 5 × 7.

2. Find the number of factors of 120.

Answer: factors

Correct! 120 = 2³ × 3 × 5, so factors = (3+1)(1+1)(1+1) = 16.

3. State the divisibility rule of 8.

A number is divisible by 8 if its last 3 digits are divisible by 8its last digit is 8.

Excellent! Check the last three digits of the number.

4. Write the smallest number divisible by 3, 4, and 5.

Answer:

Perfect! LCM(3, 4, 5) = 60.

5. What is the HCF of 24, 36, and 60?

Answer:

Great! HCF(24, 36, 60) = 12.

Drag each problem type to its solution method:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Using divisibility rules, find whether 91845 is divisible by 9 and 11.

For 9: Sum of digits = 9 + 1 + 8 + 4 + 5 =

27 is divisible by 9 →

For 11: Odd places: 5 + 8 + 9 =

Even places: 4 + 1 =

Difference = 22 - 5 =

17 is not divisible by 11 →

Excellent! 91845 is divisible by 9 but not by 11.

2. Find the LCM and HCF of 45 and 60 by prime factorization method.

45 = × = 9 × 5

60 = × ×

HCF = × =

LCM = × × =

Perfect! HCF = 15, LCM = 180.

3. If a number leaves remainder 3 when divided by 7, what will be the remainder when it is divided by 14?

Let the number = 7k + 3

When divided by 14: If k is even, remainder = 310

If k is odd, remainder = 103

Great! Remainder can be 3 or 10 depending on the number.

4. Find the smallest 4-digit number divisible by 8, 12, and 15.

LCM(8, 12, 15) =

Smallest 4-digit number = 1000

1000 ÷ 120 = 8 remainder

Next multiple = 120 × 9 =

Excellent! The smallest 4-digit number is 1080.

5. Using divisibility rules, check if 27335 is divisible by 5, 7, and 11.

For 5: Last digit = →

For 11: (5+3+7) - (3+2) = - =

10 is not divisible by 11 →

For 7: (Requires actual division or advanced method)

Great! 27335 is divisible by 5 but not by 11.

Part A: Section C – Long Answer Questions (4 Marks Each)

1. Write the divisibility test for 7 and verify whether 67228 is divisible by 7 or not.

Divisibility rule for 7:

Remove last digit, double it, subtract from remaining number

Repeat until you get a small number

Testing 67228:

Step 1: 6722 - (8 × 2) = 6722 - =

Step 2: 670 - (6 × 2) = 670 - =

Step 3: 65 - (8 × 2) = 65 - =

49 = × 7, so

Excellent! 67228 is divisible by 7.

2. A number is divisible by 2, 3, and 5. What is the smallest 4-digit number satisfying this?

LCM(2, 3, 5) =

Smallest 4-digit number =

1000 ÷ 30 = 33 remainder

Next multiple = 30 × 34 =

Perfect! The smallest 4-digit number is 1020.

3. Find the greatest 5-digit number divisible by 8, 12, and 18.

LCM(8, 12, 18) =

Greatest 5-digit number =

99999 ÷ 72 = 1388 remainder

Greatest 5-digit multiple = 99999 - 63 =

Excellent! The greatest 5-digit number is 99936.

4. A number is divisible by 4, 5, and 9. Find the least 4-digit number satisfying the condition.

LCM(4, 5, 9) =

Smallest 4-digit number = 1000

1000 ÷ 180 = 5 remainder

Next multiple = 180 × 6 =

Perfect! The least 4-digit number is 1080.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. Which of the following numbers is divisible by both 3 and 9?

(a) 423 (b) 639 (c) 712 (d) 455

Perfect! 639: Sum = 6+3+9 = 18 (÷9), so divisible by both 3 and 9.

7. The remainder when 456 is divided by 7 is:

(a) 0 (b) 1 (c) 2 (d) 3

Correct! 456 ÷ 7 = 65 remainder 2.

8. 2⁴ × 3³ × 5² has how many factors?

(a) 60 (b) 45 (c) 50 (d) 40

Excellent! Number of factors = (4+1)(3+1)(2+1) = 5 × 4 × 3 = 60.

9. The smallest prime number greater than 50 is:

(a) 51 (b) 53 (c) 55 (d) 59

Perfect! 53 is prime (51 = 3×17, 55 = 5×11).

10. The number 4620 is divisible by which of the following?

(a) 6 and 8 (b) 5 and 9 (c) 4 and 9 (d) 2, 3, and 5

Correct! 4620: Even (÷2), ends in 0 (÷5), sum = 12 (÷3). So divisible by 2, 3, and 5.

🎉 Exceptional Achievement! You've Mastered Advanced Number Theory!

Here's what you learned:

• Advanced Divisibility Rules:

  Divisibility by 7 (one method):

  • Remove last digit, double it, subtract from remaining number
  • Repeat until small number
  • Example: 343 → 34 - (3×2) = 28 → 2 - (8×2) = -14 (÷7) ✓

  Divisibility by 11:

  • Alternating sum: (sum of odd place digits) - (sum of even place digits)
  • Result must be 0 or divisible by 11
  • Example: 2728 → (8+7) - (2+2) = 15 - 4 = 11 ✓

  Divisibility by 8:

  • Last 3 digits must be divisible by 8
  • Example: 1000 → 000 ÷ 8 = 0 ✓

• Factor Counting Formula:

  • If n = p₁^a × p₂^b × p₃^c × ...
  • Number of factors = (a+1)(b+1)(c+1)...
  • Example: 120 = 2³ × 3 × 5
    • Factors = (3+1)(1+1)(1+1) = 4 × 2 × 2 = 16 factors

• Complex LCM and HCF Problems:

  Finding smallest/largest n-digit numbers:

  1. Find LCM of given numbers
  2. For smallest: Divide smallest n-digit by LCM, add remainder
  3. For largest: Divide largest n-digit by LCM, subtract remainder

  Example: Smallest 4-digit ÷ by 8, 12, 15

  • LCM = 120
  • 1000 ÷ 120 = 8 rem 40
  • Answer: 120 × 9 = 1080

• HCF × LCM Formula:

  • HCF(a,b) × LCM(a,b) = a × b
  • Useful for finding one when other is known
  • Example: HCF(45,60) = 15, so LCM = (45×60)÷15 = 180

• Prime Factorization Methods:

  1. Factor Tree Method
  2. Continuous Division Method
  3. Both give same result

  Example: 315 = 3² × 5 × 7

• Problem-Solving Strategies:

  Type 1: Smallest number divisible by a, b, c

  • Find LCM(a, b, c)
  • For n-digit: Find next multiple after smallest n-digit number

  Type 2: Greatest number divisible by a, b, c

  • Find LCM(a, b, c)
  • For n-digit: Find last multiple before largest n-digit number

  Type 3: Remainder problems

  • If remainder is r when divided by d
  • Number = d × k + r (where k is quotient)

  Type 4: Testing multiple divisibility

  • Check each divisibility rule systematically
  • Use prime factorization when helpful

• Advanced Concepts:

  • Co-prime numbers: HCF = 1
  • Perfect numbers: Sum of factors = 2n
  • Abundant/Deficient numbers
  • Twin primes: Prime pairs differing by 2 (11-13, 17-19)

• Quick Checks:

  • Divisible by 6: Must be divisible by both 2 and 3
  • Divisible by 12: Must be divisible by both 3 and 4
  • Divisible by 15: Must be divisible by both 3 and 5
  • Divisible by 18: Must be divisible by both 2 and 9

These advanced number theory concepts are crucial for competitive mathematics, cryptography, and computer science!