The question asks for the length of the longest rod that can exactly measure three given lengths: 5 m 13 cm, 11 m 34 cm, and 12 m 15 cm. To find the longest rod that can exactly measure these lengths, we need to find the Greatest Common Divisor (GCD) of these lengths.

First, we must express all lengths in the same unit. The smallest unit used is centimeters, so let's convert all lengths to centimeters.

• 1 meter (m) = 100 centimeters (cm)

Converting the given lengths to centimeters:

• 5 m 13 cm = \( (5 \times 100) \text{ cm} + 13 \text{ cm} = 500 \text{ cm} + 13 \text{ cm} = 513 \text{ cm} \)
• 11 m 34 cm = \( (11 \times 100) \text{ cm} + 34 \text{ cm} = 1100 \text{ cm} + 34 \text{ cm} = 1134 \text{ cm} \)
• 12 m 15 cm = \( (12 \times 100) \text{ cm} + 15 \text{ cm} = 1200 \text{ cm} + 15 \text{ cm} = 1215 \text{ cm} \)

Now, we need to find the GCD of 513, 1134, and 1215. We can use the prime factorization method to find the GCD.

Let's find the prime factors of each number:

• Prime factorization of 513:

  \( 513 \div 3 = 171 \)

  \( 171 \div 3 = 57 \)

  \( 57 \div 3 = 19 \)

  \( 19 \div 19 = 1 \)

  So, \( 513 = 3 \times 3 \times 3 \times 19 = 3^3 \times 19 \)

• Prime factorization of 1134:

  \( 1134 \div 2 = 567 \)

  \( 567 \div 3 = 189 \)

  \( 189 \div 3 = 63 \)

  \( 63 \div 3 = 21 \)

  \( 21 \div 3 = 7 \)

  \( 7 \div 7 = 1 \)

  So, \( 1134 = 2 \times 3 \times 3 \times 3 \times 3 \times 7 = 2 \times 3^4 \times 7 \)

• Prime factorization of 1215:

  \( 1215 \div 5 = 243 \)

  \( 243 \div 3 = 81 \)

  \( 81 \div 3 = 27 \)

  \( 27 \div 3 = 9 \)

  \( 9 \div 3 = 3 \)

  \( 3 \div 3 = 1 \)

  So, \( 1215 = 3 \times 3 \times 3 \times 3 \times 5 = 3^4 \times 5 \)

Now, let's look at the prime factorizations to find the common factors and their lowest powers:

• \( 513 = 3^3 \times 19 \)
• \( 1134 = 2 \times 3^4 \times 7 \)
• \( 1215 = 3^4 \times 5 \)

The only common prime factor among 513, 1134, and 1215 is 3.

The powers of 3 in the factorizations are \(3^3\), \(3^4\), and \(3^4\). The lowest power of the common factor 3 is \(3^3\).

Therefore, the GCD of 513, 1134, and 1215 is \(3^3\).

\( \text{GCD}(513, 1134, 1215) = 3^3 = 3 \times 3 \times 3 = 27 \)

The GCD is 27 cm.

The length of the longest rod which can be used to measure exactly the lengths 5 m 13 cm, 11 m 34 cm, and 12 m 15 cm is 27 cm.

Revision Table: Key Concepts for Measuring Lengths

Additional Information: Understanding GCD and Measurement Problems

The problem of finding the longest rod that can measure given lengths exactly is a classic application of the Greatest Common Divisor (GCD). When a rod of a certain length can measure a longer length exactly, it means the longer length is a multiple of the rod's length. If a single rod can measure multiple lengths exactly, then each of those multiple lengths must be a multiple of the rod's length. The longest such rod will have a length that is the largest common factor of all the lengths being measured.

Steps for Solving Such Problems:

1. Ensure all given measurements are in the same unit. Convert units if necessary (e.g., meters and centimeters to just centimeters).
2. Find the prime factorization of each length (in the common unit).
3. Identify the common prime factors among all the lengths.
4. For each common prime factor, take the lowest power that appears in any of the factorizations.
5. Multiply these lowest powers of the common prime factors together. The result is the GCD.
6. The GCD is the length of the longest rod in the chosen unit.

Understanding unit conversion is crucial in such problems. For example, converting 5 m 13 cm to centimeters involves knowing that 5 meters is 500 centimeters, and then adding the remaining 13 centimeters.