Question

X' is a number formed by writing the first 1002 natu-
ral numbers one after another from left to right then
find the remainder when X is divided by 9 is​

Answer 1

Answer:

the number will be like

1234567891011121314.................10011002

we know from divisibility rule that when a number is divided by 9 the remainder is same when the sum of digits of the number is divided by 9. Thus remainder in this case will be same as remainder in case when

1+2+3+4+5+6+7+8+9+10+11+12+........+1001+1002

is divided by 9.

we know that sum of first n natural number is

n(n+1)/2

so sum of first 1002 natural number will be

1002(1002+1)/2

= 501*1003

= 502503

again applying the same rule, the remainder will be same when 5+0+2+5+0+3 is divided by 9

now

15 = 6 mod 9

hence remainder is 6.

(note, we could have reapplied the rule before multiplying the number 501 and 1003.

in that case (5+0+1)*(1+0+0+3) = 6*4 = 24 when divided by 9 leaves remainder 6)