Notes on Divisibility and Parity (Problems 1–5)

Problem 1: 9x^2 + 6x + 3 is divisible by 3

Statement: The expression is divisible by 3 for all integers x.
Factorization approach: 9x^2 + 6x + 3 = 3(3x^2 + 2x + 1)
- Since the entire expression is 3 times another integer-valued polynomial, it is a multiple of 3 for all integer x.
Modular check (alternative):
9x^2 \equiv 0 \pmod{3}, \\ 6x \equiv 0 \pmod{3}, \\ 3 \equiv 0 \pmod{3}
so 9x^2 + 6x + 3 \equiv 0 \pmod{3}.
General principle: If all coefficients share a common factor d, then the polynomial value at any integer x is divisible by d (as seen with 3 here).
Quick verification example: If x = 2, the value is 9(4) + 6(2) + 3 = 36 + 12 + 3 = 51, and 51 is divisible by 3.
Connection to foundational ideas:
- Common factor extraction
- Divisibility by a fixed integer regardless of x
- Use of modular arithmetic as a quick check

Expression factoring:
8x^2y + 4xy^2 = 4xy(2x + y)
Conclusion: The expression has a factor of 4, i.e., it is divisible by 4 for all integers x, y. Therefore it is always even (and in fact always a multiple of 4).
Why this guarantees evenness:
- If a number is a multiple of 4, it is certainly divisible by 2.
Example: x = 1, y = 1 gives 8(1)^2(1) + 4(1)(1)^2 = 8 + 4 = 12, which is even.
Notes:
- Even stronger: the expression is a multiple of 4 for all integer x, y.
- The garbled transcript snippets (e.g., "-72 (anything)" or "@2 (4 xy ) + 2(xy ³)") can be ignored in favor of the clear factorization above.

Key parity fact: 4x is even, so 4x + 1 is odd for any integer x.
Therefore, the square of an odd number is odd:
(4x + 1)^2 ext{ is odd for all integers } x.
Expansion check: (4x + 1)^2 = 16x^2 + 8x + 1
- The first two terms, 16x^2 and 8x, are even; adding 1 yields an odd number.
Examples:
- If x = 0: (4(0) + 1)^2 = 1 (odd)
- If x = 1: (4(1) + 1)^2 = 25 (odd)
Additional perspective:
- Modular property: (4x + 1)^2 \equiv 1 \pmod{2}
- More broadly, any odd number squared remains odd.

Factorization:
16x^2 - 24x + 8 = 8(2x^2 - 3x + 1)
Since 8 is a factor, the entire expression is a multiple of 8 for all integers x.
Optional factoring: 2x^2 - 3x + 1 = (2x - 1)(x - 1)
- So another compact form is:
  16x^2 - 24x + 8 = 8(2x - 1)(x - 1).
Example: x = 3 → 16(9) - 72 + 8 = 144 - 72 + 8 = 80, which is divisible by 8.
Connection to earlier problems:
- Demonstrates how factoring out a common multiple ensures divisibility by that multiple.

Note: The transcript includes a statement that this is a multiple of 8 but does not provide the actual expression.
Guidance for verification if the expression is known:
- If the expression can be factored as a multiple of 8, i.e., 8k with k an integer-valued expression, then it is divisible by 8.
- Alternatively, check modulo 8 directly, or factor out an 8 if possible.
Action item: Please provide the exact fifth expression to complete the note for Problem 5.

Divisibility by a fixed integer via factoring:
- If an expression can be written as d times another integer-valued expression, it is divisible by d for all integers x.
Parity (even/odd) reasoning:
- A number is even if divisible by 2; odd if not.
- Squares preserve parity: even^2 is even; odd^2 is odd.
- If a linear expression is of the form 4x + 1, it is always odd because 4x is even.
Factoring strategies:
- Factor common factors across terms to reveal divisibility (e.g., 3, 4, 8).
- Factor quadratics to expose products that include the divisor (e.g., 8(2x - 1)(x - 1)).
Modular arithmetic quick checks:
- Modulo 3 or modulo 2 checks can confirm divisibility without full expansion.
Real-world relevance:
- These checks underpin error detection, hashing, and algorithmic simplifications where divisibility properties reduce computation.