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

Let's solve each of the problems step by step, focusing on HCF (Highest Common Factor) and LCM (Least Common Multiple) concepts.

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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.

#### Understanding:
We need a rope that can be divided exactly into pieces of 45 cm, 75 cm, and 81 cm — meaning the total length must be divisible by all three. So we are looking for the LCM of these numbers.

#### Step 1: Prime factorization
- $ 45 = 3^2 \times 5 $
- $ 75 = 3 \times 5^2 $
- $ 81 = 3^4 $

#### Step 2: LCM
Take the highest power of each prime:
- $ 3^4 $ (from 81)
- $ 5^2 $ (from 75)

So,
$$
\text{LCM} = 3^4 \times 5^2 = 81 \times 25 = 2025
$$

✔ Answer: The least length is 2025 cm.

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Problem 2:

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

#### Step 1: Find LCM of 40, 48, 60

Prime factorizations:
- $ 40 = 2^3 \times 5 $
- $ 48 = 2^4 \times 3 $
- $ 60 = 2^2 \times 3 \times 5 $

LCM = Highest powers:
- $ 2^4 $, $ 3^1 $, $ 5^1 $
$$
\text{LCM} = 16 \times 3 \times 5 = 240
$$

Now, find the largest 4-digit number divisible by 240.

Largest 4-digit number = 9999

Divide 9999 by 240:
$$
9999 \div 240 = 41.6625
$$
Take integer part: 41

Now,
$$
240 \times 41 = 9840
$$

✔ Answer: The greatest 4-digit number divisible by 40, 48, and 60 is 9840.

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Problem 3:

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

This means the total number of saplings must be divisible by 12, 15, and 40.

So we need LCM of 12, 15, 40

Factorize:
- $ 12 = 2^2 \times 3 $
- $ 15 = 3 \times 5 $
- $ 40 = 2^3 \times 5 $

LCM:
- $ 2^3 $, $ 3 $, $ 5 $
$$
\text{LCM} = 8 \times 3 \times 5 = 120
$$

✔ Answer: The least number of saplings is 120.

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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?

We want to pack all fruits equally in cartons such that:
- Each carton has same number of oranges, apples, pears.
- No fruit is left over.
- We want maximum number of cartons, so each carton should have minimum number of fruits (but same across types).

So, we need to divide each fruit type equally among cartons.

Let the number of cartons be $ x $. Then:
- $ x $ divides 210, 252, and 294.

To maximize $ x $, we need the GCD (HCF) of 210, 252, and 294.

#### Step 1: Find HCF of 210, 252, 294

Use Euclidean algorithm:

First, HCF of 210 and 252:
- $ 252 - 210 = 42 $
- $ 210 \div 42 = 5 $ → remainder 0 → HCF = 42

Now HCF of 42 and 294:
- $ 294 \div 42 = 7 $ → remainder 0 → HCF = 42

✔ So, HCF = 42

Thus, maximum number of cartons = 42

Check:
- Oranges per carton: $ 210 / 42 = 5 $
- Apples: $ 252 / 42 = 6 $
- Pears: $ 294 / 42 = 7 $

All integers → valid.

✔ Answer: The biggest possible number of cartons is 42.

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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.

Let the number be $ N $. Then:
- $ N \equiv 3 \mod 9 $
- $ N \equiv 6 \mod 12 $
- $ N \equiv 18 \mod 24 $
- $ N \equiv 39 \mod 45 $

We observe:
- $ 3 = 9 - 6 $
- $ 6 = 12 - 6 $
- $ 18 = 24 - 6 $
- $ 39 = 45 - 6 $

So, in all cases, the remainder is 6 less than the divisor.

That is:
- $ N + 6 $ is divisible by 9, 12, 24, and 45.

So, $ N + 6 $ is a common multiple of 9, 12, 24, 45.

We need the largest 5-digit number $ N $ such that $ N + 6 $ is divisible by LCM of 9, 12, 24, 45.

#### Step 1: Find LCM of 9, 12, 24, 45

Prime factors:
- $ 9 = 3^2 $
- $ 12 = 2^2 \times 3 $
- $ 24 = 2^3 \times 3 $
- $ 45 = 3^2 \times 5 $

LCM = $ 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360 $

So, $ N + 6 = 360k $ → $ N = 360k - 6 $

We want the largest 5-digit $ N $, i.e., $ N < 100000 $

So,
$$
360k - 6 < 100000 \\
360k < 100006 \\
k < 100006 / 360 \approx 277.794
$$

So maximum $ k = 277 $

Then,
$$
N = 360 \times 277 - 6 = 99720 - 6 = 99714
$$

✔ Check:
- $ 99714 \div 9 = 11079 \times 9 = 99711 $, remainder = 3 ✔
- $ 99714 \div 12 = 8309 \times 12 = 99708 $, rem = 6 ✔
- $ 99714 \div 24 = 4154 \times 24 = 99696 $, rem = 18 ✔
- $ 99714 \div 45 = 2215 \times 45 = 99675 $, rem = 39 ✔

Perfect!

✔ Answer: The greatest 5-digit number is 99714.

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✔ Final Answers:

1. 2025 cm
2. 9840
3. 120
4. 42 cartons
5. 99714

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