A number theory problem by Akshat Sharda
Find smallest positive integer such that it is a multiple of 9 and it has no odd digits.
The answer is 288.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
4 solutions
Moderator note:
Nice standard approach.
Bonus question : Can you find the second smallest integer that satisfy the same criteria?
What about 0?
Log in to reply
"positive non-zero"
Log in to reply
in question it is only written 'positive integer'
In ques it is mentioned positive no. So 0 can't be the ans... If it would have mentioned non negetive then the ans could be 0
same question here- why not zero?
Log in to reply
Because 0 is a neutral number. It is neither positive nor negative. Therefore 0 isn't a POSITIVE INTEGER
468 would be second? And 486 third? Then 648, 666.... Woah! I loved this sum of digits thing! Had never heard about this. Does this kind of sum of digits rules apply to some other number multiples as well?
Log in to reply
It is also true for 3, but no other numbers have this property. And the reason is quite simple and surprising!
Let's say we have a 5-digit number a b c d e . We can write this number as a × 1 0 4 + b × 1 0 3 + c × 1 0 2 + d × 1 0 1 + e × 1 0 0
Which equals a × ( 1 0 4 − 1 ) + b × ( 1 0 3 − 1 ) + c × ( 1 0 2 − 1 ) + d × ( 1 0 1 − 1 ) + a + b + c + d + e
And since all numbers in the form 1 0 n − 1 are divisible by 9, all the numbers with those coefficients must be divisible by 9 as well, hence we can see if this number is divisible by 9 just by checking if a + b + c + d + e is (simple modular arithmetics :>)
If you understood the concept, you can see why this works with 3 as well; because it divides 9. Though there are no other numbers that have this property, well, except 1 but it already divides every integer so it's not worth mentioning.
(Edited to fix a simple but stupid mistake)
Second smallest integer that meets the conditions is 468. Since 288+18=306, we need to look at numbers starting with 4, so the next two digits have to add up to 14. What's the smallest 2nd digit that will lead to a single digit for the 3rd? 6 leading to 8.
Odd digits means it cannot have any odd digit 18 has 1 which is odd😆😆😆
468 . I wrote a script to solve that :P
'288' '468' '486' '648' '666' '684' '828' '846' '864' '882'
can u explain which divisibility rule that led you that " all the digits have to add up to at least 18 if they are all even " ?! Thanks !
But 1 is an odd number...
Log in to reply
18 is just the digits' sum and the question says nothing about the sum of the digits having to contain just even numbers.
Log in to reply
Ohh...Got it, Thanks @Luke Johnson-Davies for clearing my doubt.
If the digits add up to 9, at least one of the digits will always be odd. If the digits add up to 18, it is possible for all the digits to be even. For 18, let's try: 99 (no), 189 (no), 288 (yes).
Hence 288 is the answer.
2+8+8=18, So the 288 is the smallest +ve intiger with no odd digits
468 is the second number like this
Using divisibility rules for 9, all of the digits have to add up to at least 18 if they are all even, so we end up with 2 + 8 + 8 = 18 as digits of the smallest number.