Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.

Options

A. 7
B. 13
C. 62
D. 31

Explanation

To find the maximum capacity of a container that can measure the diesel from three tankers containing 403 litres, 434 litres, and 465 litres of diesel respectively, we need to determine the highest common factor (HCF) of these three numbers. Here’s how we can do this step by step:

### Step 1: Prime Factorization of Each Number

1. **Prime Factorization of 403:**
- Check divisibility by prime numbers:
- Not divisible by 2 (403 is odd).
- Not divisible by 3 (sum of digits 4+0+3=7 is not divisible by 3).
- Not divisible by 5 (does not end in 0 or 5).
- Check 7: 403 ÷ 7 = 57.57 (not divisible).
- Check 11: 403 ÷ 11 = 36.64 (not divisible).
- Check 13: 403 ÷ 13 = 31 (403 = 13 × 31).
- So, the prime factorization of 403 is **13 × 31**.

2. **Prime Factorization of 434:**
- Check divisibility:
- 434 is even, so divisible by 2: 434 ÷ 2 = 217.
- Check 217 for prime factors:
- Not divisible by 3.
- Not divisible by 5.
- 217 ÷ 7 = 31 (217 = 7 × 31).
- So, the prime factorization of 434 is **2 × 7 × 31**.

3. **Prime Factorization of 465:**
- Check divisibility:
- Not divisible by 2.
- 465 ÷ 3 = 155 (465 = 3 × 155).
- Check 155 for prime factors:
- Not divisible by 3.
- 155 ÷ 5 = 31 (155 = 5 × 31).
- So, the prime factorization of 465 is **3 × 5 × 31**.

### Step 2: Identify Common Factors

Now we have the prime factorizations:
- 403 = 13 × 31
- 434 = 2 × 7 × 31
- 465 = 3 × 5 × 31

The common prime factor in all three numbers is **31**.

### Step 3: Conclusion

The highest common factor (HCF) of 403, 434, and 465 is **31**. Therefore, the maximum capacity of a container that can measure the diesel from all three tankers an exact number of times is **31 litres**.

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