The number A = 200420052006....2040 is formed by putting the consecutive integers from 2004 to 2040 together.?

Question

What is the remainder when A is divided by 9?

Answer 1

A neat property of the number 9 is if you add up the digits in a number it tells you the remainder if you were to divide by 9.

(This is also why an exact multiple of 9 has digits that add up to 9 with no remainder.)

You have 37 numbers.
The thousands digits will be 2, 2, 2, --> 2 x 37 = 74
The remainder after dividing by 9 is 2

Hundreds digits are all zero.

Tens digits are 0,0,0...1,1,1,1,1,1,1,1,1,1,2,2,2,2,...,2,2,3,3,3,....,3,3,4
That's ten 1s, ten 2s, ten 3s and a 4. 10+20+30+4 = 64
The remainder after dividing by 9 is 1

Units digits are:
4, 5, 6, 7, 8, 9, 0, 1, 2, ..., etc., 7, 8, 9, 0

That would be 1 through 9 (45) four times, except subtract 1,2,3 (or 6). The remainder after dividing by 9 is 3.

Taking these remainders (2+0+1+3) you get 6.

The answer is:
The remainder is 6

Edit: To double-check the answer, lets just add 2004 through 2040.

2000 x 37 + sum(4 through 40)
74000 + 20 x 41 - 6
74000 + 820 - 6
74814

Adding the digits:
7+4+8+1+4 = 24
2+4 = 6

Remainder of 6

Answer 2

1 + 2 + 3 +... + n = n(n+1)/2

k + (k+1) + ... + n
= 1 + 2 + ... + n - (1 + 2 + ... + k-1)
= n(n+1)/2 - (k-1)k/2

4 + 5 + ... + 40
= 40(41)/2 - (3)4/2
= 820 - 6
= 814

Sum of the digits:
= 2*37 + 814 = 888 = 9*98 + 6

Remainder is 6.

If you have a TI-89 or higher, you can check this by typing in:

remain( sum( (2040 - i)10^(4i), i, 0, 36) , 9 )

Answer 3

sum of digits
= 2*37 + 10(1+2+3) + 4*45 - (1+2+3) + 4
= 74 + 60 + 180 - 6 + 4
= 312

answer
= 312 mod 9
= 6 mod 9
= 6

Answer 4

6

Answer 5

~~~
If the sum of the digits of a number is divisible by 9 , only then the number is divisible by 9
20042005....
= 2 * 37 + (4+5... 40)
74 + (37(8+ (37-1)1)/2) S(n) = n/2(2a + (n-1)d) where n = 37 (from 4 to 40) , a = 4 and d = term difference = 1 )

74+ (37(44/2)
74+ (37*22)
74+814 =
888
888 is not divisble by 9
Highest divisible number by 9 = 882
888 - 882 = 6
therefore reminder is 6
~~~

Answer 6

You have to find the sum of the digits first.
The sum is 312. 312=9*34+6
Thus, the remainder is 6.