### Solutions to Number Theory Questions - Remainders

Have given below the solutions to the questions on remainders. The questions can be found here.

1. What are the last two digits of the number 7

^{45 }?

The last two digits of 7

^{1}are 07

The last two digits of 7

The last two digits of 7

The last two digits of 7

The last two digits of powers of 7 go in a cycle - 07,49,43,01

So, the last two digits of 745 are 07

^{2}are 49The last two digits of 7

^{3}are 43The last two digits of 7

^{4}are 01The last two digits of powers of 7 go in a cycle - 07,49,43,01

So, the last two digits of 745 are 07

2. What is the remainder when we divide 3

^{90}+ 5^{90}by 34?3

^{90}+ 5

^{90 }can be written as (3

^{2})

^{45}+ ( 5

^{2})

^{45}

= (9)^{45} + (25)^{45}

Any number of the form a^{n} + b^{n} is a multiple of (a + b) whenever n is odd

So (9)^{45} + (25)^{45} is a multiple of 9 +25 = 34

So, the remainder when we divide (3^{2})^{45} + ( 5^{2})^{45 } by 34 is equal to 0

3. N

^{2 }leaves a remainder of 1 when divided by 24. What are the possible remainders we can get if we divide N by 12?This again is a question that we need to solve by trial and error. Clearly, N is an odd number. So, the remainder when we divide N by 24 has to be odd.

If the remainder when we divide N by 24 = 1, then N

^{2 }also has a remainder of 1. we can also see that if the remainder when we divide N by 24 is -1, then N

^{2 }a remainder of 1

When remainder when we divide N by 24 is ±3, then N

^{2 }has a remainder of 9

When remainder when we divide N by 24 is ±5, then N

^{2 }has a remainder of 1

When remainder when we divide N by 24 is ±7, then N

^{2 }has a remainder of 1

When remainder when we divide N by 24 is ±9, then N

^{2 }has a remainder of 9

When remainder when we divide N by 24 is ±11, then N

^{2 }has a remainder of 1

So, the remainder when we divide N by 24 could be ±1, ±5, ±7, or ±11

Or, the possible remainders when we divide N by 24 are 1, 5, 7, 11, 13, 17, 19, 23

Or, the possible remainders when we divide N by 12 are 1, 5, 7, 11

Labels: CAT number theory, CAT Solutions, Remainders

## 0 Comments:

Post a Comment

Subscribe to Post Comments [Atom]

<< Home