Prime factors of numbers

Question

Without using a calculator find four prime factors of 3^32-2^32 which are less than 100

Answer 1

3³² - 2³² = (3¹⁶)² - (2¹⁶)²

= (3¹⁶ - 2¹⁶)(3¹⁶ + 2¹⁶)

= [(3⁸)² - (2⁸)²](3¹⁶ + 2¹⁶)

= (3⁸ - 2⁸)(3⁸ + 2⁸)(3¹⁶ + 2¹⁶)

= (3⁴ - 2⁴)(3⁴ + 2⁴)(3⁸ + 2⁸)(3¹⁶ + 2¹⁶)

= (3² - 2²)(3² + 2²)(3⁴ + 2⁴)(3⁸ + 2⁸)(3¹⁶ + 2¹⁶)

= (3 - 2)(3 + 2)(3² + 2²)(3⁴ + 2⁴)(3⁸ + 2⁸)(3¹⁶ + 2¹⁶)

= (1)(5)(13)(97)(6817)(43112257)

= (1)(5)(13)(97)(17)(401)(43112257)

5, 13, 17, and 97 are prime factors of 3³² - 2³²