On diving 223^224 by 25 reminder will be

M4maths Previous Puzzles

10 Apr 2012 reminder

On diving 223^224 by 25 reminder will be

OPtion

1) 4
2) 6
3) 9
4) 12
5) 16
6) 19
7) 20
8) 21
9) 23
10) none of these

Solution

223 and 25 are co prime.
so 223^[25*{1-(1/5)}] will give 1 as reminder on dividing by 25.Here 5 is the only factor of 25.
so {223^20}^11 will give 1 as reminder on dividing by 25.
so remaining powers of 223 = 4
223 = 200+23
so we have to divide 23*23*23*23 by 25
or 529*529 by 25
or 4*4 by 25 so reminder = 16

Correct Option: 5 Best Solution (11)

Balakrishnan M   12 years AGO

(223)^224/25=(225-2)^224/25
by using binomial expansion,
(225-2)^224/25=(225^224-224*225*2+......+2^224)/25
=[(multiples of 25)+2^224]/25
=(reminder 0+2^224)/25
=(2^7)^32/25
2^7 divides 25 leaves a reminder is 3
=(3)^32/25
=(3^3)^10*3^2/25
3^3 divides 25 leaves a reminder is 2
=9*2^10/25
=(2^7*2^3*9)/25
=3*8*9/25
=27*8%25
=2*8
=16

Like? Yes (5) | No (1)  

Tijo Jose   12 years AGO

By Euler's Theorem, 223^phi(25) mod 25 = 1
phi(25) = 20
So, 223^20 mod 25 = 1
223^224 mod 25
= (223^220)mod25 * 223^4mod 25
= (223^20mod25)^11 * 223^4mod 25
= 223^4mod 25
= (223mod25)^4mod 25
= (-2)^4 mod 25
= 16 mod 25
= 16

Like? Yes (2) | No   

Sikandar chouhan   12 years AGO

223^4=2472973441
223^8=..............41
.
.
.

223^224=......................................41

so divide first term by 25 will give remainder 16
which is same for other terms ..

Like? Yes (2) | No (2)  

sumanth jayam   12 years AGO

=(223^224)%25
=(223^24)%25 as 224 % 25 = 24
Reminder will be of ( 223^ 24 ) % 25 = 16

Like? Yes (2) | No (2)  

edwin jebakumar   12 years AGO

223^224mod 25
(223^7mod 25)32
(.88*25)*32
704/25=remainder 16

Like? Yes (1) | No (1)  

Rakesh Kumar Kohli   12 years AGO

The last two digits of 223 are 23. When we will take powers of 223 we will find that after power 20 , the last two digits of 223 i.e 23 starts repeating.It means after every 20 power, the last two digits i.e 23 repeats. So power 221 will have last two digits as 23. So the last two digits of 223^224 will be the same as last two digits of 223^4 i.e 41. So when it is divided by 25, the reminder will be 16. So on dividing 223^224 by 25, reminder will be 16.

Like? Yes (2) | No (3)  

raja   12 years AGO

=> 223^224=1.047826367387420645198347729516e+526
=> 1.047826367387420645198347729516e+526\25
=> 16
=> Ans :16 (Option : 5)

Like? Yes (1) | No (3)  

J.P.Raja Suresh   12 years AGO

=> 223^224=1.047826367387420645198347729516e+526
=> 1.047826367387420645198347729516e+526\25
=> 16
=> Ans :16 (Option : 5)

Like? Yes (1) | No (3)  

Srivass   12 years AGO

(223^224)%25=(1.047826367387420645198347729516e+526)%25=16

Like? Yes (1) | No (3)  

J.P.Anitha   12 years AGO

=> 223^224=1.047826367387420645198347729516e+526
=> 1.047826367387420645198347729516e+526\25
=> 16
=> Ans :16 (Option : 5)

Like? Yes (1) | No (3)  

Trav   12 years AGO

Other than setting it up in a calculator, I have no reason. Am looking forward to reading the reason(s).

Like? Yes (1) | No (3)