[gargsaar]

Q:- Ten's digit of 32^32??

Sol:-- Find the remainder when it is divide by 100.

(32^32)/100.

100= 25 * 4 (co-prime factors)

(32^32)/25 gives u remainder as 1.
(32^32)/4 gives remaider as 0.

Now find a number that gives 1 as remainder with 25 and 0 with 4. And the number is 76.

Thus ten's digit is 7.

[naresh99]

Originally Posted by varun nakra1

Another method is write 32^32 =(30+2)^32 Now opening the binomial expression gives us the last two terms as 30*32*2^31 whch gives (....80) as a number and the last term as 2^32..Now 2^32 =[(1024)^3]*4..then 24*24 gives 76 as the last two digits then 76*24 again gives us 24 and finally 24*4 gives 96 as the last two digits (...96)..Thus the binonial expression gives us all zeroes at the unit's place and a 6 at the unit's place coz of the last term..and it gives all zeroes at the tenth place and one 9 and one 8 at the tenth place..i.e. .........00+......00+........+.......80+........96 =............76 as the answer

But i still feel modulo by 100 is a better method

32^32 = 2^(5*32) =2^160

every 2^(10*odd) ends with 24
every 2^(10*even) ends with 76