The fractional part of 2 78 31 is, 2 31, 4 31, 6 31, 8 31

The fractional part of 2 78 31 is

Question: Answered & Verified by Expert

The fractional part of

\frac{2^{78}}{31}

MathematicsBinomial TheoremJEE Main

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Solution:

1107 Upvotes Verified Answer

The correct answer is:

\frac{8}{31}

2^{78} = 2^{3} \cdot 2^{75} = 8 {(2^{5})}^{15} = 8 {(31 + 1)}^{15}

= 8 (^{15} C_{0} 3 1^{15} +^{15} C_{1} 3 1^{14} + . . . . . . . . +^{15} C_{14} 31 +^{15} C_{15})

2^{78} = 8 (31 k + 1)

\frac{2^{78}}{31} = 8 k + \frac{8}{31}

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