Mathematical Reasoning and Problem Solving
This domain checks logical structure and disciplined strategy rather than a fixed formula list: recognising the converse and contrapositive, judging whether a claim is proved, and using parity, divisibility and digit cycles to short-cut long computations.
Core rules
- A statement is a sentence that is definitely true or false. The negation of “all A are B” is “some A are not B”, not “no A is B”.
- For the implication p → q: the converse is q → p, the inverse is ¬p → ¬q, and the contrapositive is ¬q → ¬p. Only the contrapositive is logically equivalent to the original.
- One counterexample destroys a universal claim; one example never proves one.
- Divisibility tests: by 3 or 9 from the digit sum, by 4 from the last two digits, by 11 from the alternating digit sum.
- Parity argument: if n² is even then n is even; equivalently an odd n has an odd square.
- Units digits of powers of 2, 3, 7, 8 repeat with period 4.
Worked example
Find the units digit of 7^2024. The units digits of 7¹, 7², 7³, 7⁴ are 7, 9, 3, 1, then the cycle repeats. Divide the exponent by 4: 2024 = 4 × 506 with remainder 0, so the digit is the fourth in the cycle, namely 1.
Traps
- Treating the converse as if it were proved by the original statement.
- “Proving” a general claim with a single numerical example.
- Negating “some students passed” as “no student passed”.
- A remainder of 0 in a period-4 cycle means the 4th entry, not the 0th or the 1st.
- Assuming a pattern continues without stating why it must.