Both the end digits of a 999 digit number N is 3 and N is divisible by 11, then find all the middle digits.
pattern observation: are the following numbers divisible by 3 ?
363 , yes
3113, 3223, 3333, ... ,3993, yes
35013, 34573..... , yes
For any natural number N, if you add up the odd-placed digits and if
you add up the even-placed digits, and you get the same result, then
N is divisible by 11.
e.g. for 35013
1st + 3rd + 5th digits = 3+0+3 = 6
2nd + 4th = 5 + 1 = 6
for 34573
3+5+3 = 11
4+7 = 11 , yes
try a larger number, as long as the sums as described above are equal
then the number is divisible by 11
See if you can make use of that, at the moment I am drawing a blank.
Seems like an onerous task if you want N to have 999 digits
just noticed a typo in my opening sentence.
Should have been:
pattern observation: are the following numbers divisible by 11 ?
Is answer 4???
To find the middle digits of a 999-digit number, given that both end digits are 3 and the number is divisible by 11, we'll need to use the divisibility rule for 11 and some algebraic calculations. Here's how to approach the problem:
Step 1: Understand the divisibility rule for 11.
A number is divisible by 11 if the difference between the sum of its digits at odd positions and the sum of its digits at even positions is either 0 or a multiple of 11.
Step 2: Utilize the divisibility rule for 11.
Since the number is divisible by 11, the difference between the sum of the digits at odd positions and the sum of the digits at even positions must be 0 or a multiple of 11.
Step 3: Establish the pattern.
Since we know the ones digit is 3, the sum of the digits at even positions will be the sum of all the other digits in odd positions. Therefore, the sum of the digits at even positions must also have a ones digit of 3.
Step 4: Solve for the middle digits.
In order to find the middle digits of the 999-digit number, we can determine the range within which the sum of the digits at even positions would fall. By trying different possibilities, we can find the corresponding middle digit configurations.
Let's consider some possibilities:
- One possible sum of the digits at even positions is 3.
In this case, the middle digits could be any combination of digits that result in a sum of 3 at even positions. For example, the middle digits could be all 3's, or a combination such as 1, 2, and 7.
- Another possible sum of the digits at even positions is 14.
In this case, the middle digits could be any combination of digits that result in a sum of 14 at even positions. For example, the middle digits could be all 1's, or a combination such as 2, 4, 3, 1, and 4.
By systematically trying different possibilities for the sum of the digits at even positions, you can find all the configurations of middle digits that satisfy the given conditions of the problem.
Remember, these are just examples, and there could be various other combinations that satisfy the conditions.