Maths Olympiad - Try this prime-factor question

1. Find the number of prime factors of 2 + 22 + 23 + … + 215 + 216
which are distinct.

2. Find the number of prime factors of 2 + 22 + 23 + … + 215 + 216
which are distinct.
Solution:
We have
2 + 22 + 23 + 24 + 25 + 26 + 27 + … + 215 + 216
= 2 × (1 + 2 + 22
+ 23
+ 24
+ 25
+ 26
+ … + 214
+ 215
)
= 2 × ((1 + 2) + 22 × 1 + 2 + 24 × (1 + 2) + 26 × (1 + 2) + … + 214 × (1 + 2))
= 2 × 3 × (1 + 22 + 24 + 26 + 28 + 210 + 212 + 214)
= 2 × 3 × ( 1 + 22
+ 24
× 1 + 22
+ 28
× 1 + 22
+ 212
× 1 + 22
)
= 2 × 3 × 5 × (1 + 24
+ 28
+ 212
)
= 2 × 3 × 5 × ( 1 + 24 + 28 1 + 24 )
= 2 × 3 × 5 × 17 × (1 + 28)
= 2 × 3 × 5 × 17 × 257
Answer: 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 ℎ𝑎𝑠 5 𝑑𝑖𝑠𝑡𝑖𝑛𝑐𝑡 𝑝𝑟𝑖𝑚𝑒 𝑓𝑎𝑐𝑡𝑜𝑟𝑠