Question

If the fractional part of the number 2⁴⁰³/15 is (k/15),
then k is equal to: (A) 6 (B) 8
(C) 4 (D) 14
[JEE Main 2019]

Answer 1

Answer:

B - 8

Step-by-step explanation:

Consider , 2403=2400+3=8⋅2400=8⋅2400=8⋅(24)100=8⋅(16)100=8(1+15)100=8(1+100C1(15)+100C2(15)2+....+100C100(15)100)

[By binomial theorem , (1+x)n=nC0+nC1x+nC2x2+...nCnxn,n∈N]

=8+8(100C1(15)+100C2(15)2+...+100C100(15)100)

=8+8×15λ

where λ=100C1+....+100C100(15)99∈N

∴240315=8+8×15λ15=8λ+815

⇒{240315}=815

(where {⋅} is the fractional part function )

∴k=8

Alternate Method

2403=8.2400=8(16)100.

Note that, when 16 is divided by 15, gives remainder 1.

∴ When (16)100 is divided by 15, gives remainder 1100=1 and when 8(16)100 is divided by 15, gives remainder 8.

∴{240315}=815.

(where {.} is the fractional part function)

`rArr k=8

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