Question

find the number which is divisible by 19 and sum of its digit is also 19(CAT 2016)

Anyone at this app who can solve this?​

Answer 1

Concept to remember:- If any number is divided by 9 then the remainder which we will get will be eqaul to the sum of digits of that no. or it will be eqaul to the remainder obtained by dividing the sum of digits with 9.

For example:-

i)23

Sum of digits=5.....i)

Remainder when divided by 9=5......1)

So,I) =2)

1)=2)

ii)17

Sum of digits=8

Remainder when divided by 9=8

iii)47

sum of numbers =11

11divided by 9 gives remainder=2

Remainder when 47 divided by 9=2

iv)68

Sum of numbers=14

dividing 14 by 9 given remainder 5

Remainder when the number 68 will be divided by 9 is also 5

Now let's move on question

Solution:-As it is already said,sum of the number is 19,which means

remainder obtained when it will be divided by 9=remainder obtained when it will be divided by 19=1

so this no. will be in the form 9m+1

As the number is divisible by 19 so it will also be in the form 19n

we can relate

19n=9m+1

we know that 19 is the number which on divided by 9 will have remainder 1

so n have to be 1

also we know 1=9m+1

so n is also equal to 9m+1

putting this value we get

required no. =19n=19(9m+1)

now though hit and trial put the value of m as 1,2,3..and check which no. have digit as sum of 19.

putting

putting m=5 we we will get the required no.

required no. =19×(9×5+1)

=>19×46=874

Hence 874 is the required no.

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