Home
Class 12
MATHS
The last two digits of 3^400 is...

The last two digits of `3^400` is

Text Solution

Verified by Experts

We have , `3^(400) = (3^(2))^(200) = (9)^(200) = (10 -1)^(200)`
` = (10)^(200) - ""^(200)C_(1) (10)^(199) + ""^(200)C_(2) (10)^(198) - ""^(200)C_(3) (10)^(197)+ ..+ ""^(200)C_(199) (10)^(2) - ""^(200)C_(199)(10) + 1`
` = 100 mu = ""^(200)C_(199) (10) + 1 " where" mu in I `
` 100 mu - ""^(200)C_(1) (10) + 1 = 100 mu - 2000 + 1`
` = 100 (mu - 20 ) + 1 = 100 p + 1 ` , where p is an integer
Hence , the last two digits of ` 3^(400) " is " 00 + 1 = 01 ` .
Promotional Banner

Similar Questions

Explore conceptually related problems

The last two digits of the numbers 3^(400) are 01.

While dialing a telephone number, an old man forgets its last two digit. If the last two digits are distinct, then the probability that a man dials true number is ......

What is the last digit of 6^(100) .

A four digit number (numbered from 0000 to 9999) is said to be lucky if sum of its first two digits is equal to the sum of its last two digits. If a four digit number is picked up at random then the probability that it is lucky number is :-

The number of 4-digit numbers that can be made with the digits 1,2,3,4 and 5 in which atleast two digits are identical, is

The sum of digits of a two digit number is 10. The number obtaiend by interchanging the digits is 36 more than the original number. Find the original number.

Given below are two pairs of statements. Combine these two statements using if and only if. p: If the sum of digits of a number is divisible by 3, then the number is divisible by 3. q:If a number is divisible by 3, then the sum of its digits is divisible by 3.

In a certain city, all telephone numbers have six digits , the first two digits always being 41 or 42 or 46 or 60 or 64. How many telephone number have all six digits distinct ?

Numberse are selected at random, one at a time, from the two-digit numbers 00,01,02,….99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, find probability that the event E occurs at least 3 times.

If M is even prime number,N is least prime number of two digits & P is greatest prime number of two digits, then (P+N)/(M) is dividible by-