m is a natural number. m/260 has a remainder of 60. What is the remainder of m/26 ?

Question

m is a natural number. m/260 has a remainder of 60. What is the remainder of m/26 ?

 

I can't seem to find a theorem to help me out with this question. I have two variables!

Answer 1

m/260 = q + 60/260

 

multiplying the above fact by 10 gives m/26 = 10q + 60/26.

 

Observe that the last term is 2 8/26 so the whole number part ( 2) can be incorporated (added) into the quotient 10q, giving 8 as the remainder.

Answer 2

"m is a natural number"   means

 

"A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. The set of natural numbers, denoted N, can be defined in either of two ways:
N = {0, 1, 2, 3, ...}
N = {1, 2, 3, 4, ...}"             

 

If m/260 has a remainder of 60 then:

   m/(2*2*5*13) = n remainder 2*2*3*5      [use prime factors;  the quotient, n, is also a natural number]

So, m = n*(2*2*5*13) + (2*2*3*5)         [multiply both sides by 2*2*5*13]

      m = 2*13*(n*2*5) + 2*2*3*5            [factor out 26 to get ready]

 

Now, divide m by 26 (for m/26) to determine the remainder (note: we don't care about the quotient):

m/26 = m/(2*13)

         = ( 2*13*(n*2*5) + 2*2*3*5) ) / (2*13)

         = (n*2*5)  +  (2*2*3*5)/(2*13)

         = (n*2*5) + 60/26

         = (n*2*5) + 2   with remainder 8

We are only interested in the reminder:  the remainder is 8

 

Try a case:  let m=5*260+60 = 1360

   m/260 = 5 with remainder 60          [this fits problem description]

  What is the remainder of m/26 ?

     1360/26 = 52 with remainder 8

Answer 3

m/260 has a remainder of 60

260+60=320

320/260=1 R 60

m=320 (there are other values for m; such as 580 for example))

320/26=12 R 8

Answer 4

Since the remainder when m is divided by 260 is 60, m = 260k + 60, for some integer, k.  k is the quotient.

 

m = 260k + 60 = 26(10k) + 26 + 26 + 8

 

                       = 26(10k) + 2(26) + 8

 

                       = 26(10k+2) + 8    Since k is an integer, so is 10k+2.

 

If m is divided by 26, the divisor is 26, the quotient is 10k+2, and the remainder is 8.