Question

1..Two milk containers contains 398 l and 436 l of milk. The milk is to be transferred to another container with the help of a drum. While transferring to another container 7l and 11l of milk is left in both the containers respectively.What will be the maximum capacity of the drum.
2. Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer, q.
3.Find HCF of 56, 96 and 324 by Euclid’s algorithm.

Answer 1

volume V1 = 398 liters
V2 = 436 Litres
     a drum of capacity D litres is used to transfer the milk to other containers.  When we transfer milk, we transfer D litres, 2D litres, 3 D , 4D etc. amounts.  If there is any milk left out, that will be a fraction of the capacity of the drum.

  let m and n be integers.
   V1 = 398 = m * D + 7      =>    m * D = 391 = 17 * 23
   V2 = 436 = n * D + 11    =>    n * D = 425  =  5 * 5 * 17
        Looking at both equations above,  D = 17  , as only 17 occurs in both.

answer is 17 liters.
==============================
2.
      n is a positive integer.
     n * n = 5 q or 5q + 1  or  5q + 4      for some  integer  q.

   any integer number n  can be expressed as   in one of the following 5 forms.
            n =      5 k,   5 k + 1,    5 k + 2    or    5 k +3   or    5 k + 4

  so obtain squares of each of the above forms.

SO,    n * n =
      25 k²    =  5 ( 5 k²)    or,                                    so  here q = 5 k²
      25 k² + 10 k + 1    = 5 (5k²+ 2k) + 1    or,        so  here q = 5 k² + 2k
      25 k² +  20 k + 4  = 5 (5 k² + 4 k) + 4         or,      here q = 5 k² + 4 k
      25 k² + 30 k + 9  =  5 (5 k² + 6k + 1) + 4  or
      25 k² + 40 k + 16  =  5 (5k² + 8k + 3) + 1               

=========================
3.   
     HCF(324, 96)
   324 = 96 * 3 + 36
   9 6  =  36 * 2 + 24
   36 =  24 * 1 + 12
   24 = 12 * 2 + 0          =>  12 is the HCF

  HCF(56, 12)
     56 = 12 * 4 + 8
     12 = 8 * 1 + 4
     8 = 4 * 2 + 0      =>  4 is the HCF

hence 4 is the answer.

verify :  324 = 9 * 9 * 4
           56 = 2 * 7 * 4
           96 = 2 * 2 * 3 * 2  * 4