sums of digits that are divisible by 10 - samllest consecutive numbers

[Minkynb]

Hi all

Pease help - I am 49 yrs old so this is not homework cheating

I need to find the smallest 2 conescutive natural numbers where the sum of digits are divisible by 10

I would really appreciate any assistance

Nick

 

[Ishuda]

Hi all

Pease help - I am 49 yrs old so this is not homework cheating

I need to find the smallest 2 conescutive natural numbers where the sum of digits are divisible by 10

I would really appreciate any assistance

Nick

Assuming conescutive is supposed to be consecutive: Any two consecutive digits can be written as (a)2n and 2n+1 or (b)2m+1 and 2m+2. Is the sum of the two consecutive digits odd or even? Is 10 odd or even?

EDIT: It has been pointed out to me very gently, thanks (possible) descendant of Genghis, that maybe what you meant was something like the following:
a=13; sum of digits = 4
b=14; sum of digits = 5
4+5 = 9
So, 13 and 14 doesn't work. Are there any two consecutive numbers which will produce 'a sum of digits' number divisible by 10? Let
a = x₀ + 10 x₁ + 10² x₂ + ... + 10ⁿ x_n
b = y₀ + 10 y₁ + 10² y₂ + ... + 10ⁿ y_n
where b=a+1. If A is the sum of the digits of a, and B is the sum of the digits of B, then if x₀ is eight of less, B = A+1, for example if x₀ is 0, then y₀ is 1 and y_j=x_j for j=1, 2, 3, ..., n; if x₀ is 1, then y₀ is 2 and .... So if x₀ is 8 or less the sum of digits of the two numbers is odd. For 10 to divide that number it must be divisible by 2 and 5 and an odd number is not divisible by 2. So the conclusion is, if the sum of digits is to be divisible by 10, x₀ must be 9.
...

 

[Ishuda]

9,10 : sum digits = 10
59,60 : sum digits = 20
149,150 : sum digits = 20

So, in each case the smaller number has an 'x₀' of nine. Taking the 'each number is two digits case', since the sum of any two digits must be less than 20 and the sum of all digits must add to a number divisible by 10, then y₁=x₁+1 and either y₁+x₁=1 or y₁+x₁=1. So 2x₁+1=1 or 2x₁+1=11. Obviously it must be the case of the latter, i.e. two digit numbers means neither x₁ nor y₁ is 0, so x₁=5 and y₁=6, i.e. 59, 60.

Do a similar process with 'one 1 digit and a two digit' [obviously x₀=9 or if we let a two digit number have a leading digit of zero, then x₀ above is 0], two 3 digit numbers, etc. [or build a computer program to spit them out].