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.

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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

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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