Aptitude - Numbers - Discussion

Can you please tell the answer for this?

A number x when divided by 9 leaves a remainder 8. What is the remainder when we divide x^24 by 9?

A. 3
B. 2
C. 7
D. 1

Here 29 is fit for this by hit and trail method when 29/4 give reminder 1 and when 29/5 gives reminder 4 so why 4, 1 is not answer.

Answer should be 4 and 1.

Let a number X which is divisible by 4 and 5 respective gives 1 and 4.

Step 1:

Now, first divide by 4. We take Y as dividend. When you divide X by 4 it gives remainder 1.

4 | X | Y.
______

1.

So X = 4 * Y + 1 --- equation(1).

Step 2:

Now, divide by 5. We take 1 as dividend because it divides completely. When you divide Y by 5 it gives only remainder 4.

5 | Y | 1.
______

4.

So Y = 5*1+4.

Step 3:

Y = 9.

And put this value in equation (1).

X = 4*9+1.
X = 37.

Step 4:

Now 37 divide by 5 and 4 respective it will gives 2 and 3.

When dividing a positive integer n by another positive integer D (divider) , we obtain a quotient Q, which is a non-negative integer and a remainder R, which is an integer such that 0\leq{R}<D. We can write n = DQ+R.

When dividing our number n by 4 we obtain a remainder of 1, so, if the quotient is some integer Q, we can write n = 4Q+1.

Now, dividing Q by 5, we obtain another quotient say q and remainder 4, thus we can write
Q = 5q+4.

It follows that n = 4(5q+4)+1 = 20q+17.

Since n = 20q+17 = 5(4q+3)+2, it means that when dividing n by 5 first, we get a quotient 4q+3 and remainder 2.

Then dividing 4q+3 by 4 we obviously obtain a remainder of 3.

Divident=9, and next time it divided by 7 and 6. (not 5 and 4).

Let a no x.

It is divided by 4 remainder is 1.

So x = 4*y + 1;
y = 5*z+ 4;( anther division).

Now,
x=4(5*z +4)+1= 20*z+17;

We can write above no.as
x=5(4*z +3)+2;

So above no gives remainder 2 when divided by 5. And then successive division gives remainder 3 from ( 4*z +3).

Let the number be X. when it is divided by 4, remainder is 1 and quotient is Y(assume). now further when Y is divided by 5, remainder is 4 and quotient will be 1(successively divided). Now according to formula,

Y=(5*1)+4=9 and
X=(4*y)+1=(4*9)+1=37

So we got the values of X and Y..rest you can find out easily.

Earlier I mentioned (Y-1) which will be only (Y), that is when (X) is divided by 4 the result is (Y).

It is said that "divided SUCCESSIVELY by 4 and 5" .
SUCCESSIVE: Following one another
So from the given solution we see that at first any number (X) has been divided by 4 and then the result (Y-1) is divided by 5 .