when a 4 digit positive integer x is divided by 13 the remainder is 7 what is x

Question

 when x is divided by 12 the remainder is 7 when x is divided by 11 the remainder is 7 when x is divided by 10 the average is 7 what is the value of x

Answer 1

Reworded:  “When a 4 digit positive integer x is divided by 13 the remainder is 7;  when x is divided by 12 the remainder is 7; when x is divided by 11 the remainder is 7; when x is divided by 10 the remainder is 7; what is the value of x?”
 
First think about remainders, dividends, divisors, and quotients:
     A dividend divided by a divisor produces a quotient with a remainder (using integer arithmetic).
     That means that:  (the quotient)*(the divisor) + (the remainder) = (the dividend)
 
So, if a dividend x produces:
     Remainder of 7 when x is divided by 13,    then (x-7) must be evenly divisible by 13.

         [note:    ( x/13 ) = quotient  with  remainder 7  ]

     Remainder of 7 when x is divided by 12,    then (x-7) must be evenly divisible by 12.
     Remainder of 7 when x is divided by 11,    then (x-7) must be evenly divisible by 11

     Remainder of 7 when x is divided by 10,    then (x-7) must be evenly divisible by 10.

 

The prime factors are:

      13 = 13

      12 = 2 * 2 * 3

      11 = 11

      10 = 2 * 5

 

To be evenly divisible by 13, 12, 11, and 10, the number (x+7) must be a multiple of 13*2*2*3*11*5 = 8580.  [note: this is called the LCM (least common multiple) of 13, 12, 11, and 10 – choose only as many factors as needed to include the number using the given factors (for example, 2 appears twice which is the most times that it appears in any factor)].

 

To have a remainder of 7 when divided by 13, 12, 11, and 10, the value x must be a (multiple of 8580) + 7.  The lowest of these values is 8587 (and it is the only 4-digit one).

    x = 8587

 

Check:

      8587/13 = 660   rem 7

      8587/12 = 715   rem 7

      8587/11 = 780   rem 7

      8587/10 = 858   rem 7  

Answer 2

Lets convert this to a division problem using long division.

 

     ___

13 | x

 

But in order to find x, we need to go backwards.  We multiply the divisor by a possible quotient, c, then add the result to 7.  The following expression that represents this value of x is

 

x = 13(c) + 7

 

 

Pick a number for c that is positive and multiply it by 13.  Then add 7 to the result.  You should get a 4 digit number.

 

x = 13(76) + 7 = 988 + 7 = 995

x = 13(77) + 7 = 1001 + 7 = 1008

 

1008 is the least 4 digit number.

 

 

Try this process for the rest of the problems you have posted here.