Question

Prove that 25^11 - 5^19 is divisible by 31.

Answer 1

See below

Explanation:

25¹¹ - 5¹⁹

Convert to base 5 = (5²)¹¹ - 5¹⁹

= 5²² - 5¹⁹

Factor out the GCF = 5¹⁹(5³ - 1)

= 5¹⁹(125 - 1)

= 5¹⁹ × 124

= 5¹⁹ × 4 × 31

31 is a factor of 25¹¹ - 5¹⁹, so the expression is divisible by 31.