Recall the following properties :

a³ - b³ = (a - b)(a² + ab + b²)
a³ + b³ = (a + b)(a² - ab + b²)
a² - b² = (a - b)(a + b)
(aᵐ)ⁿ = aᵐⁿ

So, 2²⁴ - 1

= (2¹²)² - 1²

= (2¹² - 1)(2¹² + 1)

= ((2⁴)³ - 1³)((2⁴)³ + 1³)

= (2⁴ - 1)((2⁴)² + 2⁴(1) + 1²)(2⁴ + 1)((2⁴)² - 2⁴(1) + 1²)

= (16 - 1)(16² + 16 + 1)(16 + 1)(16² - 16 + 1)

= 15(256 + 16 + 1)(17)(256 - 16 + 1)

= 3(5)(273)(17)(241)

= 3(5)(3)(91)(17)(241)

= 3²(5)(7)(13)(17)(241)

So, prime factorisation of 2²⁴ - 1 is 3² × 5 × 7 × 13 × 17 × 241

There are 7 prime factors, of which 6 are unique.

Next, how many factors are there?


We can split the prime factors into cases of two groups :


① One group has 1 prime factor, the other group has the other 6 prime factors.

Number of ways to choose 1 prime factor for the first group (the rest goes into the second group) = 6 (since there are 6 unique factors only)

Number of unique factors here = 6 × 2 = 12
(Because you have 2 groups for each way, so that's two distinct factors per way.


② One group has 2 unique prime factors, the other 5 goes into the second group.

Number of ways to choose 2 unique prime factors for the first group
= 6C2
= 6! ÷ 2! ÷ (6 - 2)!
= 6! ÷ 2! ÷ 4!
= 6 × 5 ÷ 2!
= 30 ÷ 2
= 15

Number of unique factors here = 15 × 2 = 30


③ One group has 3 unique prime factors, the other 4 goes into the second group.

Number of ways to choose 3 unique prime factors for the first group
= 6C3
= 6! ÷ 3! ÷ (6 - 2)!
= 6! ÷ 3! ÷ 3!
= 6 × 5 × 4 ÷ 3!
= 5 × 4
= 20

Number of unique factors = 20 × 2 = 40

④ One group has two 3's, the other group has the other 5 unique primes. (1 way only, so 2 factors here).

⑤ One group has two 3's and another unique prime factor. The other group has the remaining 4 unique primes.

Number of ways = 5C1 = 5
Number of factors here = 5 × 2 = 10

⑥ the number 2²⁴ - 1 itself and 1
Number of factors = 2


Total

= 12 + 30 + 40 + 2 + 10 + 2 = 96

Alternative way to factorise (longer)

2²⁴ - 1

= (2⁸)³ - 1³

= (2⁸ - 1)((2⁸)² + (2⁸)(1) + 1²)

= ((2⁴)² - 1²)((2⁸)² + 2(2⁸)(1) + 1² - 2⁸)

(Completing the square)

= (2⁴ - 1)(2⁴ + 1)((2⁸ + 1)² - (2⁴)²)

= ((2²)² - 1²)(16 + 1)(2⁸ + 1 + 2⁴)(2⁸ + 1 - 2⁴)

= (2² - 1)(2² + 1)(17)((2⁴)² + 2(2⁴)(1) + 1 - 2⁴)(256 + 1 - 16)

(Completing the square again)

= 3(5)(17)((2⁴ + 1)² - (2²)²)(241)

= 3(5)(17)(2⁴ + 1 + 2²)(2⁴ + 1 - 2²)(241)

= 3(5)(17)((2²)² + 2(2²) + 1 - 2²)(13)(241)

(A third time)

= 3(5)(17)((2² + 1)² - 2²)(13)(241)

= 3(5)(17)(2² + 1 + 2)(2² + 1 - 2)(13)(241)

= 3(5)(17)(7)(3)(13)(241)

= 3²(5)(7)(13)(17)(241)


2²⁴ - 1 = 3² × 5 × 7 × 13 × 17 × 241