IIT JEE , AIEEE Forum - Mathematics , Algebra

umang

Here r a few questions -

1. What is the remainder when 4 to the power 96 is divided by 6 ???

2. 4(4!)+5(5!)+........20(20!) when divided by 64 , what is the remainder ?????

Note : Pls post complete solutions

gcch29

1)if observe we will find that we get same remainder if wether we divide a no. with 3 or 6
4⁹⁶=2¹⁹²
=(3-1)¹⁹²
=¹⁹²C₀3⁹⁶+¹⁹²C₁3¹⁹¹(-1)¹.....................+¹⁹²C₁₉₂

=3N+1 , N belongs to natural no

remainder=1

2)_{[ 0]}

^{[n ]} r.(r!)=(r+1)!-1

=21!-24

=64N-24 , N belongs to natural no.

remainder 64-24=40

gcch29

1)if observe we will find that we get same remainder if wether we divide a no. with 3 or 6
4⁹⁶=2¹⁹²
=(3-1)¹⁹²
=¹⁹²C₀3⁹⁶+¹⁹²C₁3¹⁹¹(-1)¹.....................+¹⁹²C₁₉₂

=3N+1 , N belongs to natural no

remainder=1

2)_{[ 0]}

^{[n ]} r.(r!)=(n+1)!-1

=21!-24

=64N-24 , N belongs to natural no.

remainder 64-24=40

umang

Hey gcch29 ,

I thought the first que. the other way and got 4 as the answer . See =

4^1=4

4^2=16

4^3=64

4^4=256

4^5=1024

4^6=4096

.

..

...

and so on .

Just see the last digit , and notice the trend . even powers end at 6 and odd ones end at 4 . So , 4^96 will end at 6 . Now , divide 16 or 256 or 4096 by 6 , and u get remainder as 4 . So , it follows that remainder will be 4 .

umang

and , the options I hav for the 2nd question r 24,36,60 and 12 .

So , thats wrong .

umang

Hey !

Pls reply !!!!

gcch29

hey umang do u find any mistake in my soln.
i have tried many times i still got the same result

lazycol

Q1:

4⁹⁶ = 2¹⁹² = 2*2¹⁹¹ = 2(3-1)¹⁹¹

= 2(3¹⁹¹ - ¹⁹¹C₁3¹⁹⁰ + ¹⁹¹C₂3¹⁸⁹ -...+¹⁹¹C₁₉₀3 - 1)

= 2*3N -2 = 6N -2 =6(N-1) +4

SO REMINDER IS 4

Q2:

i agree with GCCH29.

ans is definitly 40,ther is sumthin wrong in ur opts

lazycol

in Q1 gcch argued dat reminder when a no is divided by 3&6 r same

it isnt true always

reminder when divided by 6 is either same as dat when divided by 3 or 3+dat value

take d eg of 5

5 divided by 3 gives 2 as reminder & 6 gives 3+2 ie 5

Aatish

gr8 work lazycol.........

pushml

4^{96₌₄6_*44_*44}

^now

^{4^6=4(mod 6),}

^{4^4=4(mod 6),}

^so

4⁹⁶⁼⁴*⁴*^{4(mod 6)=64(mod 6)=4(mod 6)}

^{so the ans for Q 1 IS 4.PLS RATE ME IF U THINK THIS IS CORECCT}

umang

Hey !

The answer given for 2nd que is 60 .

rohan151190

HEY,

FOR THE 1ST QUE. REMAINDER IS 0

pushml

hey rohan,prove your point!!!!!!!

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