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 7⁴⁵ ?

The last two digits of 7¹ are 07

The last two digits of 7²are 49
The last two digits of 7³ are 43
The last two digits of 7⁴ are 01

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. What is the remainder when we divide 3⁹⁰ + 5⁹⁰ by 34?

3⁹⁰ + 5⁹⁰ can be written as (3²)⁴⁵ + ( 5²)⁴⁵

= (9)⁴⁵ + (25)⁴⁵

Any number of the form aⁿ + bⁿ 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 (3²)⁴⁵ + ( 5²)⁴⁵ by 34 is equal to 0

3. N² 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 N² also has a remainder of 1. we can also see that if the remainder when we divide N by 24 is -1, then N² a remainder of 1

When remainder when we divide N by 24 is ±3, then N² has a remainder of 9
When remainder when we divide N by 24 is ±5, then N² has a remainder of 1
When remainder when we divide N by 24 is ±7, then N² has a remainder of 1
When remainder when we divide N by 24 is ±9, then N² has a remainder of 9
When remainder when we divide N by 24 is ±11, then N² 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