The exponent of $$3$$ in $$100$$! is

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Infinity Learn · Accepted Answer

Now,$$100$$!$$=1$$⋅$$2$$⋅$$3$$⋅…$$98$$⋅$$99$$⋅$$100=(1$$⋅$$2$$⋅$$4$$⋅$$5$$…$$98$$⋅$$100)(3$$⋅$$6$$⋅$$9$$⋅.$$.96$$⋅$$99)=K$$⋅$$333(1$$⋅$$2$$⋅$$3$$…$$32$$⋅$$33)$$[∵ let $$K=1$$⋅$$2$$⋅$$4$$⋅$$5$$…$$98$$⋅$$100$$]$$=$$[$$K(1$$⋅$$2$$⋅$$4$$…$$31$$⋅$$32)$$]$$333$$⋅(3$$⋅$$9$$⋅$$12$$…$$30$$⋅$$33)=K1$$⋅$$333$$⋅$$311(1$$⋅$$2$$⋅$$3$$…$$10$$⋅$$11)∵ let $$K(1$$⋅$$2$$⋅$$4$$…$$31$$⋅$$32)=K1=K1(1$$⋅$$2$$⋅$$4$$…$$10$$⋅$$11)333$$⋅$$311(3$$⋅$$6$$⋅$$9$$⋅$$12)=K2333$$⋅$$311$$⋅$$34(1$$⋅$$2$$⋅$$3$$⋅$$4)=K3$$⋅$$333$$⋅$$311$$⋅$$34$$⋅$$3$$ ∵ let $$K2(1$$⋅$$2$$⋅$$3$$⋅$$4)=K3=K3$$⋅$$349Hence$$, exponent of $$3$$ is $$49$$

The exponent of 3 in 100! is

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