Testing Cases
Summary
Even-odd ACT problems can be solved either by testing all four parity cases for two integers or by using parity logic to reason more efficiently. The strongest approach is to use the four-case method as a learning tool and then build toward direct logical analysis for faster, deeper understanding.
- Start with the expression p^2 + pq and test all four possibilities for p and q: both even, both odd, p even with q odd, and p odd with q even.
- Only the case where p is odd and q is even makes p^2 + pq odd, so that combination must be true.
- Use logic to confirm the same result: if p were even, both terms would be even; if p is odd, then pq must be even for the sum to stay odd, which forces q to be even.
- Apply the same four-case method to 4p + q and observe that the working cases are exactly the ones where q is odd.
- Use logic to see why p does not matter in 4p + q: 4p is always even, so the sum is odd only when q is odd.
- Build number sense by starting with the methodical four-case approach, but always revisit the problem with logic so parity reasoning becomes quicker and more natural.
Chapters
00:00
Using the Four-Case Method for Even-Odd Problems
01:57
Solving the Same Parity Problem with Logic
04:54
Building Number Sense Beyond Rote Case Testing