If x = 2³² − 1, then what is the sum of the factors of x that lie between 1…
2026
If x = 2³² − 1, then what is the sum of the factors of x that lie between 1 and 100?
Hint: 2^(2ⁿ) + 1 is prime for n = 1, 2, 3, 4.
- A.
314
- B.
176
- C.
143
- D.
431
Attempted by 233 students.
Show answer & explanation
Correct answer: B
We need to find the factors of x = 2³² − 1 that lie between 1 and 100.
Use the factorization:
2³² − 1 = (2¹ + 1)(2² + 1)(2⁴ + 1)(2⁸ + 1)(2¹⁶ + 1)
So,
2³² − 1 = 3 × 5 × 17 × 257 × 65537.
Only the prime factors 3, 5, and 17 can form factors below 100. The factors between 1 and 100 are:
3, 5, 15, 17, 51, 85
Sum = 3 + 5 + 15 + 17 + 51 + 85 = 176.
Correct Answer: Option B (176).