Write the natural numbers 1,2,3, ,998,999 as a number N 1234 998999, and find the sum of the digits

And so on 10 99=55+65+75+85+95+105+...135=855

And so on 100 999 1000+1100....1800=12600

We will n=1234....The 998999 is divided into 10 sections.

0 1 2……99 (0 is added here to compare with the following part) we can see.

The next 9 parts are all the first part, and the first part is preceded by hundreds (1-9), which are 100, 1,100, and 2 ......, respectively100 9s (100......199 is a number of 100).

Then just. n = x 10 +100+200+……900x 10 + 4500

The same method of analysis.

It is. So it is.

x 10 +10+20+……90

So. n = x 10 + 4500

900 x 10 +4500

The sum of the digits of both n is 13500

This uses the knowledge of sequences.

The 、... of the sequenceSum of n-2, n-1, n = n*(n+1) 2

Specifically, this question is the sum = 999 * (999 + 1) 2 = 499500

And so on 10 99=55+65+75+85+95+105+...135=855

And so on 100 999 1000+1100....1800=12600

The sum of all the numbers should be the sum of the 1,000 natural numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Single digits: 0 to 9 occur 100 times each, and the sum is (0+1+2+...)

8 + 9) * 100 = 4500 decimal digits: 0 appears 91 times, 1 to 9 each appears 100 times, and the sum is 0 * 91 + (1 + 2 + ...8 + 9) * 100 = 4500 hundred answer orange:

0 appears 1 time, 1 to 9 each appears 100 times, and the sum is 0*1+(1+2+....)8+9)*100=4500 thousands: 1 appears 1 time, and the sum is 1*1=1 So, the sum is 4500 + 4500 + 4500 + 1 = 13501

Summary. We can solve this problem using the sequence summation formula. The continuous natural number 1 809 can be seen as a series of equal differences, the first term is 1, and the tolerance is 1, so the formula for summation of the sequence can be used:

s n = frac $$ where $s n$ is the sum of the first $n$ terms, $a 1$ is the first term, and $a n$ is the $n$ term. Substitute the formula into the calculation. First of all, there are a total of 809 terms in this equal difference series, the first term $a 1 = 1$, and the last term $a = 809$

s = frac = 328,405 $$ Therefore, the sum of all the numbers of the consecutive natural number 1 809 is 328,405.

Find the sum of all the numbers of 1 809, which is a continuous natural number.

We can solve this problem using the sequence summation formula. The continuous natural number 1 809 can be regarded as a series of equal differences, the first term is 1, and the tolerance is 1, so Li Infiltration can use the formula for summation of the series: $$s n = frac $$, where $s n$ is the sum of the first $n$ terms, $a 1$ is the first term, and $a n$ is the $n$ term.

Substitute the formula into the calculation. First of all, there are a total of 809 terms in this equal difference series, the first term $a 1 = 1$, and the last term $a = 809$, which is substituted into the formula to get the following: $$s =frac = 328,405 $$ Therefore, the sum of all the numbers of the continuous natural number 1 and 809 is 328,405.

is the sum of all the natural numbers.

It can be solved using Gauss's formula, i.e., the sum of all the numbers is equal to the first term plus the last term multiplied by the number of terms divided by 2, where the first term is 1, the last term is 809, and the number of terms is 809, so there is: the sum of all empty band numbers = 1 + 809) 809 2 = 327,615. Thus, the sum of all the numbers of 1 809, which is judged to be 809 consecutive key numbers, is 327,615. This one.

It is the sum of numbers, not the sum of earthly socks.

You can see all of them back to four digits, such as 1 as 0001, and then add 1 0000 then 0,1,2,3,9 appear 1000 times each.

So digital and (0+1+2+..)+9)*1000

Repeat the natural numbers 1, 2, 3, 4, 5, and 6 in order to get a multi-digit 123456123456....Makes up a 988-digit number. Does this number contain a factor of 3? Is it a multiple of 2?

988=6x164+4

The last four of this number are 1234

This number is a multiple of 2.

1+2+3+4+5+6=21 is a multiple of 3.

1+2+3+4=10 is not a multiple of 3.

This number does not contain a factor of 3

988 6 = 164---4, so this number is 123456123456---1234561234, and the mantissa of this number is 4, which must be a multiple of 2.

According to the characteristics of the number divided by 3, the individual digits are added and divisible by 3. The sum of 123456---123456 must be divisible by 3, and the sum of 1234 is 10 and is not divisible by 3, so this number does not contain a factor of 3.

988=6*164+4

So multiple digits are 123456123456.....1234561234, it is obviously a multiple of 2.

Because 1234 does not contain a factor of 3

So 123456123456.....1234561234 does not contain a factor of 3

987 divisible by 3.

So 988 digits are not divisible by 3, and the last digit is 4, which is divisible by 2

is a multiple of 2 and does not contain a factor of 3

丨丨! My family is all right, are we eating, good morning, dear dear, good morning, when you sleep, what are you going to do without me? Are you coming back to eat, are you dreaming, ah, don't want the system, a lot of money, how many numbers in a month, how small is the boss who left home, and there is no one, my family's ** This baby:

No, are you in your hometown? You're a**ahhh

Solution: 988 984 4

6x164＋4

So finally. The 4 digits are 1234, and the sum of the 123456 digits is 21, and the sum of the 988 digits is 21x984 10

20674 is not divisible by 3, so this number does not contain the factor 3, and the single digit of this number is 4, so it is a multiple of 2.

The teacher said to use short division, so this is not very useful, so let's say so.