verify the following statements

Question

Verify the following statements
a) 100 ≡ 5 (mod 5)
b) 112 ≡ 4 (mod 7)
c) 80≡ 8 (mod 2)

 

im so lost

Answer 1

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus.

 

-WIKIPEDIA

 

So for example. If the modulus is 6...

 

The number 9 would work as follows

1  2  3  4  5  6  7  8  9  <-- counting 1 through 9

1  2  3  4  5  6  1  2  3  <-- wrap around at modulus

So, 9 is the same is 3, if the modulus is 6.

Another quicker way to get a value is to divide and observe the remainder.

9 / 6 has a remainder of 3

 

a) 100 ≡ 5 (mod 5)  states that 100 is the same as 5 if the modulus is 5

100 / 5 has a remainder of 0

5 / 5 has a remainder of 0

They are the same (True)

 

b)

112 with modulus 7 is 0

4 with modulus 7 is 4

These are not the same (False)

 

c)

80 mod 2 is 0

8 mod 2 is 0

These are the same (True)