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Show that 18^n can never end …

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Show that 18^n can never end with the digit zero

    • Posted by Shanthi Menon 6 years, 6 months ago

    CBSE > Class 10 > Mathematics
    • 1 answers

    Sia ? 6 years, 6 months ago

    We have to find out that whether 18n can end with the digit 0 or 5 for any natural number n
    If any number ends with the digit 0 then its factors must be in the form : 2m {tex} \times {/tex} 5n
    If any number ends with the digit 5 then its factors must be in form : 5m {tex} \times {/tex} 3n or 5m {tex} \times {/tex} 7n 
    So for any number to end with the digit 0 or 5  its factors must be in form 5m {tex} \times {/tex} 3n or 5m {tex} \times {/tex} 7n  or 2m {tex} \times {/tex} 5n (where m and n are positive integers)
    Now, 18 = 2 × 9 = 2 ×  3 ×  3 = 2 ×  32
    So, 18n = (2 ×  32)n  = 2n ×  (32)n= 2n ×  32n
    18n is not in the form of  5m ×   3n or  5m ×  7n  or 2m ×  5n 
    So, 18n cannot end with 0 or 5 for any natural number n.

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