CAT4MBA

Q2) (121212......300 Times)/99 ? What is the remainder?

Solution:

If I want to find the remainder when a number is divided by 99, then I start from the unit's digit of the number, take two digits at a time and then move towards the leftmost digit finding the sum of all those two digit numbers obtained and then find the remainder when that sum is divided by 99.

Example:

What is the remainder when the number 1234567 is divided by 99?

The we first start from the right hand side of the number taking two digits at a time and find their sum.

So, we get: 67 + 45 + 23 + 01 = 136.

Now, when 136 is divided by 99, the remainder is 37.

So, the remainder when 1234567 is divided by 99, the remainder is 37.

We use the similar method for finding the remainder when the number 121212......300 times is divided by 99.

Note that if we take two digits at a time, starting from the right hand side, then everytime we will get the number 12 and since the given number is a 300 digit number, 12 will be added 150 times and so the sum obtained will be (12) * (150) = 1800 and this number, when divided by 99, the remainder will be 18, which is the required remainder.

Hence, when the number 1212121212 ............. 121212 (300 times) is divided by 99, the remainder is 18.

Thank You.

Ravi Raja

Q3) What is the digital sum of 4444^4444 ? Ans- 17

Solution:

Digital Sum implies that we keep on adding the digits of the result of 4444^4444 till we get a single digit number and that single digit number is the digital sum of the given number.

So, to find the digital sum of a given number, all we have to do is find the remainder when the number is divided by 9 and the remainder will be the required answer.

So, we find the remainder when 4444^4444 is divided by 9.

4444^4444 = (4446 - 2)^4444

Hence the remainder when 4444^4444 is divided by 9, is the same as the remainder obtained when 2^4444 is divided by 9.

2^4444 = {(2^3)^1481}*(2) = (8^1481)*(2) = {(9 - 1)^1481}*(2)

Hence the required remainder is: {(-1)^1481}*(2) = (-1)(2) = - 2 + 9 = 7

Thus, the digital sum of 4444^4444 is 7.

Thank You.

Ravi Raja

ques: 5
for the question to be true the two uys have to be born on concecutive centuries. The exact centuries do not matter:
let one be born on 19x andthe other be born on 20y and let the conversation be happening on 20z .
thus:

z-y = y
z+100-x = x
eliminating z :

100 = 2(x-y)

hence the diference of the ages = 50

28 = 4 * 7
30 = 5 * 6

tus p = 4*6
thus 6p will have (4+1) * (6+1) = 35 factors

WHAT R U TRYING TO DO..??? EXPLAIN IT MORE CLEARLY...

IN Q5)  WHAT R U TRYING TO DO..??? EXPLAIN IT MORE CLEARLY...

OOPS !!! I MEANT IN Q4)...WAT R U TRYING TO DO...EXPLAIN IT MORE CLEARLY PLZZZZZZZZZ........

well, ya i know the sol i wrote didnt real mean much.
The point is let p have any number of different prime number as factors.
if p = (a^x)*(b^y)*(c^z)...
where a,b,c are prime numbers,
then the total number of divisors of p = (x+1)(y+1)(z+1)...

now muktiplying p by 2 increases one of (x+1), (y+1), (z+1)...
and muktiplying p by 3 increases another of (x+1), (y+1), (z+1)...

now its just by logic you can figure out how many different prime number factors does p have from the info given about 2p and 3p

But why you took

28 = 4 x 7

It can be 14 x 2, 1 x 28

Similarly for the other number . . . .

we wil have to look at 28 and 35 together