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For any posutve integer n Euclid …

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For any posutve integer n Euclid division lemma to prove that n cube-n by 6

    • Posted by Sehaj Brar 6 years, 10 months ago

    CBSE > Class 10 > Mathematics
    • 1 answers

    Gaurav Seth 6 years, 10 months ago

    Let :-


    a = n³ - n


    = n(n² - 1) 


    = n(n - 1)(n + 1)


    = (n - 1)n (n + 1)


    1) Now out of three (n - 1),n and (n + 1) one must be even so a is divisible by 2


    2) Also (n - 1),n and (n + 1) are three consecutive integers thus as proved a must be divisible by 3 


    From (1) and (2)


    a must be divisible by 2 × 3 = 6


    Hence n³ - n is divisible by 6 for any positive integer n

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