Worksheet on Word Problems on H.C.F. and L.C.M. |Highest Common Factor - Free Printable

Let me work through each problem step by step.

Problem 1: Find the least length of a rope which can be cut into whole number of pieces of lengths 45 cm, 75 cm and 81 cm.

This asks for the LCM of 45, 75, and 81.

- 45 = 3² × 5
- 75 = 3 × 5²
- 81 = 3⁴

LCM = 3⁴ × 5² = 81 × 25 = 2025 cm

Let me verify:
- 2025 ÷ 45 = 45 ✓
- 2025 ÷ 75 = 27 ✓
- 2025 ÷ 81 = 25 ✓

Answer: 2025 cm

Problem 2: Find the greatest number of 4-digits which is exactly divisible by 40, 48 and 60.

First find LCM of 40, 48, 60.
- 40 = 2³ × 5
- 48 = 2⁴ × 3
- 60 = 2² × 3 × 5

LCM = 2⁴ × 3 × 5 = 16 × 3 × 5 = 240

Greatest 4-digit number is 9999.
9999 ÷ 240 = 41.6625
So 41 × 240 = 9840

Verify: 9840 ÷ 40 = 246 ✓, 9840 ÷ 48 = 205 ✓, 9840 60 = 164 ✓

Answer: 9840

Problem 3: What is the least number of saplings that can be arranged in rows of 12, 15 or 40 in each row?

Find LCM of 12, 15, 40.
- 12 = 2² × 3
- 15 = 3 × 5
- 40 = 2³ × 5

LCM = 2³ × 3 × 5 = 8 × 3 × 5 = 120

Verify: 120 ÷ 12 = 10 ✓, 120 ÷ 15 = 8 ✓, 120 ÷ 40 = 3 ✓

Answer: 120

Problem 4: 210 oranges, 252 apples and 294 pears are equally packed in cartons so that no fruit is left. What is the biggest possible number of cartons needed?

This asks for HCF of 210, 252, 294.
- 210 = 2 × 3 × 5 × 7
- 252 = 2² × 3² × 7
- 294 = 2 × 3 × 7²

HCF = 2 × 3 × 7 = 42

Verify: 210 ÷ 42 = 5, 252 ÷ 42 = 6, 294 ÷ 42 = 7 ✓

Answer: 42 cartons

Problem 5: Find the greatest number of 5-digits which on being divided by 9, 12, 24 and 45 leaves 3, 6, 18 and 39 as remainders respectively.

Notice: 9 - 3 = 6, 12 - 6 = 6, 24 - 18 = 6, 45 - 39 = 6

So the number + 6 is divisible by LCM of 9, 12, 24, 45.

LCM of 9, 12, 24, 45:
- 9 = 3²
- 12 = 2² × 3
- 24 = 2³ × 3
- 45 = 3² × 5

LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360

Greatest 5-digit number is 99999.
99999 ÷ 360 = 277.775
277 × 360 = 99720

So N + 6 = 99720
N = 99720 - 6 = 99714

Verify:
- 99714 ÷ 9 = 11079 remainder 3 ✓ (99714 = 9×11079 + 3)
- 99714 ÷ 12 = 8309 remainder 6 ✓ (99714 = 12×8309 + 6)
- 99714 ÷ 24 = 4154 remainder 18 ✓ (99714 = 24×4154 + 18)
- 99714 ÷ 45 = 2215 remainder 39 ✓ (99714 = 45×2215 + 39)

Answer: 99714

Let me double-check my calculations using Python.