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Use Euclid's division lemma to show …

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Use Euclid's division lemma to show that the cube of any positive integer is of the from 9m, 9m+1 or 9m+8.

    • Posted by Aayush Kumar Chandrawanshi 5 years, 11 months ago

    CBSE > Class 10 > Mathematics
    • 3 answers

    Bhumika Agrawal 5 years, 11 months ago

    I agree with you Yogita ingle

    0Thank You

    Bhumika Agrawal 5 years, 11 months ago

    I agree with Yogita Ingle

    0Thank You

    Yogita Ingle 5 years, 11 months ago

    Let x be any positive integer. Then, it is of the form 3q or, 3q + 1 or, 3q + 2.

    So, we have the following cases :

    Case I : When x = 3q.

    then, x3 = (3q)3 = 27q3 = 9 (3q3) = 9m, where m = 3q3.

    Case II : When x = 3q + 1

    then, x3 = (3q + 1)3

    = 27q3 + 27q2 + 9q + 1

    = 9 q (3q2 + 3q + 1) + 1

    = 9m + 1, where m = q (3q2 + 3q + 1)

    Case III. When x = 3q + 2

    then, x3 = (3q + 2)3

    = 27 q3 + 54q2 + 36q + 8

    = 9q (3q2 + 6q + 4) + 8

    = 9 m + 8, where m = q (3q2 + 6q + 4)

    Hence, x3 is either of the form 9 m or 9 m + 1 or, 9 m + 8.

    2Thank You

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