when 54 is divided by the positive integer N, the quotient is 13 and the remainder is R. What is the value of R?

Question

it is an SHSAT problem and it is hard

 

Answer 1

54/N = 13 with some R

 

When thinking about remainders, let's start simple:

 

2/2 = 1 R0

3/2 = 1 R1

4/2 = 2 R0

5/2 = 2 R1

                 We observe that when dividing by 2 the remainder is either 0 or 1

                  in other words any whole number less than 2

 

Let's choose a bigger divisor:

10/5 = 2 R0

11/5 = 2 R1

12/5 = 2 R2

13/5 = 2 R3

14/5 = 2 R4

15/5 = 3 R0

               We observe that when dividing by 5 the remainder is  0,1,2,3, or 4

                OR in other words any whole number less than 5

 

Using this pattern, what do you think is true about the remainder when dividing by any positive integer?

 

R will be a whole number less than N

R<N

R∈ {0,1,2,...,N-1}

 

**recall the the WHOLE number set is the set of zero and positive integers

 

 

Now, consider when the quotient is 13. N must be less than 5, which will impact R (it must be less than 4) now consult with the other two tutor's posts! Good luck to you!

Answer 2

54/N   =   13  remainder R             where N is a positive integer

 

Let's start with the answer:   N=4 and R=2

    Notice that

             52/4 is 13 remainder 2

 but also notice that

             52/13 is 4 remainder 2

 

This is because 4*13 is 13*4.  Both values equal 52.  So, when dividing into 54, the remainder is 2.

 

Put another way:

    54 = 13*4 + 2          [the quotient is 13 and the remainder is 2]

 

So, the problem states:

    54 = 13*N + R      [N is positive integer and R<13]  What is R?

Answer 3

If 54 divided by N gives a quotient of 13 and a remainder of 2, then:

 

13N + 2 = 54

 

13N = 52

 

N = 4

 

54/4 = 13 with a remainder of 2