Question

the colored rods shown here are 1 centimeter square and of lengths 1 cm, 2 cm, 3 cm, ..., 10 cm. we say that a rod measures another rod if a number of the first rods placed end to end are the same length as the given rod. answer parts (a) through (g).

(a) which rods will measure the 9-rod?
(type whole numbers. use a comma to separate answers as needed.)

Explanation:

Define the measurement condition

The problem states that a rod of length \(L\) measures another rod of length \(M\) if a whole number of the first rods placed end-to-end equals the length of the second rod. Mathematically, this means \(L\) must be a divisor of \(M\), or equivalently, \(M\) must be a multiple of \(L\).

Using the Multiples of Numbers knowledge point

$$ L \text{ measures } M \iff M = k \cdot L \quad \text{for some } k \in \mathbb{Z}^+ $$

Find the divisors of 9

We need to find which rod lengths \(L \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) measure the 9-rod (\(M = 9\)). This requires finding all positive integer divisors of 9 within the available rod lengths.

Using the Multiples of Numbers knowledge point

$$
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$$

The divisors of 9 are 1, 3, and 9.

Answer:

(a) Which rods will measure the 9-rod?
<blank>1, 3, 9</blank> (Type whole numbers. Use a comma to separate answers as needed.)