Number at (i) = 643 – 175 = 468

Number at (ii) = 1752 – 114 = 1638

Number at (iii) = 1149 – 213 = 936

Clearly, 468, 1638 and 936 are multiples of 234 and 234 > 213.

Divisor = 234

Let the least number be x

y = 5 × 1 + 3 = 8

x = 13 × 8 + 1 = 105

On dividing 105 by 65, remainder = 40

The remainder will be same.

On dividing 9 by 6, remainder = 3

On dividing 81 by 6, remainder = 3

On dividing the given number by 119, let k be the quotient and 19 as remainder.

Then, number = 119k + 19

= 17 × 7k + 17 × 1 + 2

= 17 (7k + 1) + 2

Hence, the given number when divided by 17, gives (7k + 1) as quotient and 2 as remainder.

Number just greater than 3 which is divisible by 7 = 7

Number just smaller than 200 which is divisible by 7 = 196

Here, a = 7, a_n = 196,

d = 7, n = 8

a_n = a + (n –1)d

⇒196 = 7 + (n – 1) × 7

⇒ 

⇒ n = 27 + 1 = 28

Note : We can find the answer after dividing 200 by 7. The quotient is our answer.

Let the required number of persons be x.

According to the question, 2x² = 3042

Or

or 

Let number (dividend) be X.

∴ X = 296 × Q + 75 where Q is the quotient and can have the values 1, 2, 3 etc.

= 37 × 8 × Q + 37 × 2 + 1

= 37 (8Q + 2) + 1

Thus we see that the remainder is 1.

[Remark : When the second divisor is a factor of the first divisor, the second remainder is obtained by dividing the first remainder by the second divisor.

Hence, divide 75 by 37, the remainder is 1].

The value of (x-y) will be 18