Question

on dividing a natural number by 13 the remainder is 3 and on dividing the same number by 21 the remainder is 11 if the number lies between 500 and 600 then the remainder on dividing a number by 19 is​

Answer 1

Answer:

4

Step-by-step explanation:

Let the number = X

Given condition

500 < X < 600

X is divisible by 21 and remainder is 11

For numbers between 500 to 600, there will be only 5 such numbers.

such as, 515, 536, 557, 578, 599

Checking the 2nd condition of division by 13, for each number

515 = 507 + 8 = 39*13 + 8

536 = 533 + 3 = 41*13 + 3

557 = 546 +11 = 42*13 + 11

578 = 572 + 6 = 44*13 + 6

599 = 598 + 1 = 46*13 + 1

∴ 536 is the natural number.

Remainder on dividing the number by 19..

536÷ 19 = 28*19 + 4 = 532 + 4

Remainder = 4

Answer 2

Answer:

536

4

Step-by-step explanation:

Let say number = X

13a = X - 3

21b = X - 11

=> 13a + 3  = 21b + 11

=> 21b = 13a - 8

=> 13b + 8b = 13a - 8

=> 13b + 8b + 8  = 13a

=> 13b + 8(b+1) = 13a

=> b + 1 = 13k  

let say k = 1

=> b = 12  

X = 21b + 11 =  21 * 12 + 11 = 263 ( not between 500 -600)

k = 2

b = 25  

X = 21 * 25 + 11 = 536

Number = 536

536/19 = 532/19  + 4/19  = 28 + 4/19

so remainder = 4