Math Problem Statement

Find the number which has the following criteria i) is a factor of 36 ii) is a multiple of 3 iii) has the sum of its two digits a square number, iv) has the product of its two digits a cube.

Solution

The problem asks you to find a number that meets the following criteria:

  1. It is a factor of 36.
  2. It is a multiple of 3.
  3. The sum of its two digits is a square number.
  4. The product of its two digits is a cube.

Let's break it down:

Step 1: Find factors of 36

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Step 2: Find multiples of 3

Since the number must be a multiple of 3, it needs to be divisible by 3. From the factors of 36, the multiples of 3 are: 3, 6, 9, 12, 18.

Step 3: Check if the sum of its two digits is a square number

We now look at the two-digit multiples of 3 from the previous step. The candidates are 12, 18.

  • For 12, the sum of the digits is 1+2=31 + 2 = 3, which is not a square number.
  • For 18, the sum of the digits is 1+8=91 + 8 = 9, which is a square number (323^2).

Step 4: Check if the product of its two digits is a cube

For 18, the product of the digits is 1×8=81 \times 8 = 8, which is a cube number (232^3).

Conclusion:

The number that satisfies all four criteria is 18.

Would you like further explanations or help with similar problems? Here are some related questions for practice:

  1. What are the factors of 36 and how do you list them?
  2. How do you determine if a number is a multiple of 3?
  3. How can you check if the sum of two digits forms a square number?
  4. How do you verify if the product of two digits is a cube?
  5. What other numbers could be factors of 36 and meet certain mathematical criteria?

Tip: When solving problems that involve multiple conditions, it’s often useful to break the problem down step by step, testing each condition individually.

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Math Problem Analysis

Mathematical Concepts

Factors
Multiples
Square Numbers
Cube Numbers

Formulas

Sum of digits
Product of digits

Theorems

Factors of a number
Multiples of a number
Properties of square numbers
Properties of cube numbers

Suitable Grade Level

Grades 6-8