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8 September, 12:37

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

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  1. 8 September, 12:43
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    See below

    Step-by-step 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.
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