IMC 2004 Q20
20. What is the largest power of 2 that divides \(127^2 - 1\)?
- A \(2^1\)
- B \(2^7\)
- C \(2^8\)
- D \(2^{63}\)
- E \(2^{127}\)
Intermediate Mathematical Challenge 2004, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C \(2^8\)
Using the formula for the difference of two squares:
\(127^2 - 1 = 127^2 - 1^2 =\)
\((127 + 1)(127 - 1) = 128 \times 126 = 2^7 \times 2 \times 63 = 2^8 \times 3^2 \times 7\).
Solution © UK Mathematics Trust, from the IMC 2004 solutions.