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Find the exponent of 3 in 100!....

Find the exponent of 3 in 100!.

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In terms of prime factors 100! Can be written as `2^(a)*3^(b)*5^(c)*7^(d)`
Now, `b=E_(3)(100!)`
`=[(100)/(3)]+[(100)/(3^(4))]+[(100)/(3^(3))]+[(100)/(3^(4))]` + . . .
`=33+11+3+1+0+ . . .=48`
Hence, exponent of 3 is 48.
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Knowledge Check

  • The exponent of 3 in 33! is :

    A
    15
    B
    2
    C
    11
    D
    none of these

  • The exponent of 3 in 331 is :

    A
    a. 15
    B
    b. 2
    C
    c. 11
    D
    d. none of these

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