Moderate Level Worksheet

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

In this moderate level, we'll explore HCF, LCM, advanced divisibility rules, and prime factorization techniques.

These skills are essential for solving complex number problems and understanding number relationships.

1. Define a multiple.

A multiple is the result of multiplying a number by an integera factor of a number.

Perfect! Multiples are obtained by multiplying a number by 1, 2, 3, 4...

2. Write any two numbers divisible by both 2 and 3.

First number: 6121824 (Note: Any number divisible by 6)

Second number: 12182430 (Note: Any other number divisible by 6)

Excellent! Numbers divisible by both 2 and 3 are divisible by 6.

3. Find the number of factors of 45.

Answer: factors

Correct! Factors of 45 are: 1, 3, 5, 9, 15, 45 (6 factors).

4. Check whether 315 is divisible by 9.

Sum of digits = 3 + 1 + 5 =

9 is divisible by 9, so 315 is divisible by 9is not divisible by 9

Perfect! Since sum of digits = 9, the number 315 is divisible by 9.

5. Write all prime numbers between 10 and 20.

Prime numbers: , , ,

Excellent! These are the only prime numbers between 10 and 20.

Drag each example to its correct concept:

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

1. Write the prime factorization of 210.

210 = × × ×

Excellent! 210 = 2 × 3 × 5 × 7.

2. Find the highest common factor (HCF) of 18 and 24.

Prime factorization of 18 = ×

Prime factorization of 24 = ×

HCF = Product of lowest powers = × =

Perfect! HCF(18, 24) = 6.

3. Find the least common multiple (LCM) of 8, 12, and 16.

8 = , 12 = × , 16 =

LCM = Product of highest powers = × =

Great! LCM(8, 12, 16) = 48.

4. Check whether 756 is divisible by 6, 9, and 12.

For 6: Even? Sum of digits = 7+5+6 = (÷3) →

For 9: Sum = 18 (÷9) →

For 12: Divisible by both 3 and 4? Last 2 digits = 56 (÷4) →

Excellent! 756 is divisible by 6, 9, and 12.

5. The sum of digits of a number is 21. By which number is it divisible?

Sum = 21

21 is divisible by and

So the number is divisible by (divisibility rule: sum of digits)

Perfect! If sum of digits = 21, the number is divisible by 3.

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

1. Using divisibility rules, find whether 3240 is divisible by 3, 5, 6, and 10.

For 3: Sum of digits = 3 + 2 + 4 + 0 =

9 is divisible by 3 →

For 5: Last digit = →

For 6: Divisible by both 2 and 3? Even + sum÷3 →

For 10: Last digit = →

Excellent! 3240 is divisible by all: 3, 5, 6, and 10.

2. Write the divisibility rule of 11 and test it for 45177.

Rule: Difference between sum of digits at odd places and even places should be 0 or divisible by 11greater than 11

For 45177:

Odd places (from right): 7 + 1 + 4 =

Even places: 7 + 5 =

Difference = 12 - 12 =

Since difference = 0, 45177 is divisible by 11is not divisible by 11

Perfect! 45177 is divisible by 11.

3. Find all the factors of 72 and identify its prime factors.

Prime factorization: 72 = ×

All factors: , 2, 3, 4, 6, , , 12, 18, , 36,

Prime factors: and

Excellent! 72 has 12 factors, and its prime factors are 2 and 3.

4. A number is divisible by 3, 5, and 9. Write the smallest 3-digit number satisfying this condition.

LCM(3, 5, 9) = LCM(9, 5) = (since 9 already includes 3)

Smallest 3-digit number divisible by 45:

100 ÷ 45 = 2 remainder

Next multiple = 45 × 3 =

Perfect! The smallest 3-digit number is 135.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. The number 72 can be expressed as:

(a) 2³ × 3² (b) 2³ × 3³ (c) 2² × 3³ (d) 2³ × 3⁴

Wait, let me recalculate: 72 = 8 × 9 = 2³ × 3². The correct answer should be (a) 2³ × 3², not (c).

7. 1008 is divisible by:

(a) 4 only (b) 8 only (c) both 4 and 8 (d) neither

Correct! Last 3 digits = 008 (÷8), so divisible by both 4 and 8.

8. Which of the following is divisible by 9?

(a) 126 (b) 324 (c) 214 (d) 511

Perfect! Sum of digits: 3+2+4 = 9 (÷9), so 324 is divisible by 9.

9. Which number is divisible by 7?

(a) 161 (b) 121 (c) 141 (d) 151

Correct! 161 = 7 × 23, so it's divisible by 7.

10. The number of factors of 72 is:

(a) 12 (b) 10 (c) 9 (d) 8

Excellent! 72 = 2³ × 3². Number of factors = (3+1)(2+1) = 12.

🎉 Outstanding Work! You've Mastered Intermediate Number Theory!

Here's what you learned:

• HCF and LCM:

  • HCF (Highest Common Factor): Largest number that divides all given numbers
  • LCM (Least Common Multiple): Smallest number divisible by all given numbers
  • Formula: For two numbers a and b: HCF × LCM = a × b

• Finding HCF and LCM:

  • Prime Factorization Method:
    • HCF: Product of lowest powers of common primes
    • LCM: Product of highest powers of all primes
  • Example: 18 = 2 × 3², 24 = 2³ × 3
    • HCF = 2¹ × 3¹ = 6
    • LCM = 2³ × 3² = 72

• Advanced Divisibility Rules:

  Divisor	Rule
  4	Last 2 digits divisible by 4
  6	Divisible by both 2 and 3
  8	Last 3 digits divisible by 8
  9	Sum of digits divisible by 9
  11	Difference of (sum of odd place digits) and (sum of even place digits) = 0 or ÷11
  12	Divisible by both 3 and 4

• Prime Factorization:

  • Express number as product of prime factors
  • Example: 210 = 2 × 3 × 5 × 7
  • Use factor tree or division method

• Counting Factors:

  • If n = p₁^a × p₂^b × p₃^c
  • Number of factors = (a+1)(b+1)(c+1)
  • Example: 72 = 2³ × 3² → Factors = (3+1)(2+1) = 12

• Problem-Solving Strategies:

  1. For divisibility: Apply appropriate rule
  2. For HCF: Find common factors, take smallest powers
  3. For LCM: Find all factors, take largest powers
  4. For smallest number divisible by many: Find LCM
  5. For largest number dividing many: Find HCF

• Key Relationships:

  • If divisible by a and b, divisible by LCM(a,b)
  • Sum of digits rule works for 3 and 9
  • Last digit(s) rules work for 2, 4, 5, 8, 10

These concepts are fundamental for algebra, fractions, and advanced mathematics!