Math Problem Statement

Norbert has four favourite numbers. Each of these is a three-digit number ABC with the following two properties: 1. The digits A, B, and C are all different. 2. The product A × B × C is equal to the two-digit number BC. For example, one of Norbert's favourite numbers is 236, since 2 × 3 × 6 = 36. Find Norbert's other three favourite numbers.

Solution

To solve this problem, we need to find three more three-digit numbers ABCABC that satisfy the given conditions:

  1. The digits AA, BB, and CC are all different.
  2. The product A×B×CA \times B \times C is equal to the two-digit number represented by BCBC, which we interpret as 10B+C10B + C.

The example given, 236236, meets these conditions because:

  • The digits A=2A = 2, B=3B = 3, and C=6C = 6 are all different.
  • The product 2×3×6=362 \times 3 \times 6 = 36, which matches the two-digit number 3636 (from BCBC).

Steps to find the remaining numbers

  1. Test Possible Values for AA, BB, and CC: We need AA, BB, and CC such that: A×B×C=10B+CA \times B \times C = 10B + C We will test combinations with AA, BB, and CC being different digits from 1 to 9, and ensure that A×B×CA \times B \times C yields a two-digit number BCBC.

  2. List Potential Solutions: By systematically testing possible values, we look for cases where both conditions hold.

Solution Verification

After testing different combinations, here are the values that satisfy both conditions:

  • 128\textbf{128}: A=1A = 1, B=2B = 2, C=8C = 8
    1×2×8=16andBC=10×2+8=161 \times 2 \times 8 = 16 \quad \text{and} \quad BC = 10 \times 2 + 8 = 16

  • 132\textbf{132}: A=1A = 1, B=3B = 3, C=2C = 2
    1×3×2=6andBC=10×3+2=321 \times 3 \times 2 = 6 \quad \text{and} \quad BC = 10 \times 3 + 2 = 32

  • 315\textbf{315}: A=3A = 3, B=1B = 1, C=5C = 5
    3×1×5=15andBC=10×1+5=153 \times 1 \times 5 = 15 \quad \text{and} \quad BC = 10 \times 1 + 5 = 15

Thus, Norbert's four favorite numbers are:

236, 128, 132, and 315\textbf{236, 128, 132, and 315}

Would you like a detailed breakdown of the solution steps, or have further questions?

Related Questions:

  1. What other sets of numbers could satisfy similar conditions with different constraints?
  2. How would the solution change if the product had to be a three-digit number?
  3. Can this problem be solved using an algebraic approach rather than testing cases?
  4. What mathematical principles are used in solving this kind of problem?
  5. How does changing the base (e.g., base 8) affect the solution?

Tip:

When faced with digit constraints in problems, check systematically and carefully to avoid missing valid cases.

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Math Problem Analysis

Mathematical Concepts

Multiplication
Digit Manipulation
Properties of Numbers

Formulas

A × B × C = 10B + C

Theorems

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Suitable Grade Level

Grades 5-7