Question

Suppose b is any integer. If b mod 12 = 7, what is 8b mod 12? In other words, if division of b by 12 gives a remainder of 7, what is the remainder when 8b is divided by 12? Your solution should show that you obtain the same answer no matter what integer you start with. Using the d_

Answer 1

Answer: 8

Step-by-step explanation:

If b mod 12 gives a remainder of 7, this means that one of the integer b could be 7+12 i.e 19(remainder must always be added to any integer used) This can also be expressed as;

19/12 = 1 remainder 7

If 8b mod 12 = x where x is the remainder when 8b is divided by 12. substituting b = 19 we have;

8b/12 = 8(19)/12

= 152/12 = 12R8

Similarly using another integer say b = 24; therefore 24+7 = 31 (since the remainder which is 7 must be added to the assumed integer before it is used)

b mod 12 = 31/12

= 2R7

Substituting b = 31 in 8b mod 12 to check if we are going to arrive at the same remainder of 7 we have;

8b/12 = 8(31)/12

= 248/12

= 20R8

This shows that no matter the value of 'b' used, the remainder of 8b mod 12 will always be 8

Answer 2

Answer:

Remainder = 8

Step-by-step explanation:

1. b mod 12 = 7

Add 12 to 7

b = 7 + 12 = 19

b = 19

So, 19/12 = 1 remainder 7

The remainder when 8b mod 12 = 8b/12

= (8*19)/12 = 152/12

= 12 remainder 8

Testing with a different value of b

2. b mod 12 = 7

Add 24 (12*2) to 7

b = 7 + 24 = 31

b = 31

So, 31 mod 12 i.e., 31/12 = 2 remainder 7

The remainder when 8b mod 12 = 8b/12

= (8*31)/12 = 248/12

= 12 remainder 8 again

So, no matter the starting value of b that was obtained, 8b mod 12 will always have a remainder of 8.