Math Problem Statement

Consider multiplying a two-digit number by 11. The specific examples are: 17(11) = 187, 23(11) = 253, 41(11) = 451, 62(11) = 682. a) A conjecture states that when multiplying a two-digit number by 11, the first and last digits of the product are the digits of the original number. Is this conjecture reasonable? b) Can you find a counterexample that disproves this example? c) If this conjecture seems to be valid, can you prove it? d) If this conjecture is invalid, can you state the conditions under which it is not? Additional conjectures: a) The sum of a multiple of 3 and a multiple of 6 must be a multiple of 6. b) In any multiple of 9, the sum of the digits must also be 9. c) All natural numbers can be written as the sum of consecutive natural numbers, e.g., 9 = 4 + 5, 12 = 3 + 4 + 5.

Solution

The image you provided contains a mathematical problem involving inductive and deductive reasoning, with a conjecture related to multiplying two-digit numbers by 11. Here is a breakdown of the questions:

  1. Problem Statement:

    • Consider multiplying a two-digit number by 11. The specific examples given are:
      • 17×11=18717 \times 11 = 187
      • 23×11=25323 \times 11 = 253
      • 41×11=45141 \times 11 = 451
      • 62×11=68262 \times 11 = 682
  2. Conjecture:

    • A conjecture states that when multiplying a two-digit number by 11, the first and last digits of the product are the digits of the original number, e.g., 62×11=68262 \times 11 = 682.
    • The questions to answer are:
      • (a) Is this conjecture reasonable?
      • (b) Can you find a counterexample that disproves this example?
      • (c) If the conjecture seems to be valid, can you prove it?
      • (d) If this conjecture is invalid, can you state the conditions under which it is not valid?
  3. Additional Questions:

    • Do you agree or disagree with the following conjectures?
      • (a) The sum of a multiple of 3 and a multiple of 6 must be a multiple of 6.
      • (b) In any multiple of 9, the sum of the digits must also be 9.
      • (c) All natural numbers can be written as the sum of consecutive natural numbers (e.g., 9=4+59 = 4 + 5, 12=3+4+512 = 3 + 4 + 5).

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Math Problem Analysis

Mathematical Concepts

Inductive reasoning
Deductive reasoning
Properties of multiplication
Patterns in numbers

Formulas

Multiplication of two-digit numbers by 11
Properties of sums and divisibility

Theorems

Divisibility rules for 3, 6, and 9
Arithmetic progressions

Suitable Grade Level

Grades 10-12