QJEEM19Ma01f3: JEE Main 2019 Mathematics - Class 11 JEEM19Ma01f3 Solution: If the fractional part of the number , then k is equal to :

Question

Verified Solution • 2025–26 Session moderate

If the fractional part of the number

\left\{ {{{{2^{403}}} \over {15}}} \right\} is \, {k \over {15}}

, then k is equal to :

Accepted Answer

{{{2^{403}}} \over {15}}

= {{{2^3}\, \cdot \,{2^{400}}} \over {15}}

= {8 \over {15}}{\left( {16} \right)^{100}}

= {8 \over {15}}{\left( {15 + 1} \right)^{100}}

= {8 \over {15}}\left( {{}^{100}{C_0} + {}^{100}{C_1}\,15 + {}^{100}{C_2}{{\left( {15} \right)}^2} + .....{{\left( {15} \right)}^{100}}} \right)

= {8 \over {15}} + 8\left( {{}^{100}{C_1} + {}^{100}{C_2}\,\left( {15} \right) + ..... + {{\left( {15} \right)}^{99}}} \right)

= {8 \over {15}} + 8

(integer)

\therefore

Fractional part

= {8 \over {15}}

According to the question,

{k \over {15}} = {8 \over {15}}

\Rightarrow

K

=

8

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