Some Sum Semantics

Probability Level 4

For any natural number $x$ , let $\Upsilon (x)$ denote the sum of digits of $x$ .

Find the number of all three digit numbers such that $\Upsilon (\Upsilon (x)) = 5$ .

The answer is 100.

2 solutions

Harsh Shrivastava
Mar 29, 2015

Hint: A natural number $x$ leaves the same remainder when divided by $9$ as the remainder which is left by dividing $\Upsilon (x)$ by $9$ .

Oh! This property of 9 can really help to solve this type of question in short time.But I did it the long way solving taking cases:-(

A Former Brilliant Member - 6 years, 2 months ago

Nice property

even I had to bash this question!

Vaibhav Prasad - 6 years, 2 months ago

Yes i have seen this question..............................xD

Vaibhav Prasad - 6 years, 2 months ago

Where did you saw it?? @Vaibhav Prasad

Harsh Shrivastava - 6 years, 2 months ago

@Harsh Shrivastava – @Vaibhav Prasad RMO I think? S(S(2)) ??

Hrishik Mukherjee - 6 years, 2 months ago

@Hrishik Mukherjee – And I have noticed a very beautiful result that number of three-digit numbers that satisfy $\Upsilon (\Upsilon (x)) = 1,2,3,......,8$ are 100 .

@Vaibhav Prasad @Kalash Verma @Hrishik Mukherjee

Harsh Shrivastava - 6 years, 2 months ago

@Harsh Shrivastava – Nice observation!! :)

Hrishik Mukherjee - 6 years, 2 months ago

@Hrishik Mukherjee – Yep , I modified that problem!

Harsh Shrivastava - 6 years, 2 months ago

@Harsh Shrivastava – The name of your question suggests something =D

Hrishik Mukherjee - 6 years, 2 months ago

@Hrishik Mukherjee – what??????

Harsh Shrivastava - 6 years, 2 months ago

@Harsh Shrivastava – Like .. If people know that a same type of question had appeared in RMO then Yes, They Have Seen It :)

Hrishik Mukherjee - 6 years, 2 months ago

@Hrishik Mukherjee – Oh I see!!!!

Harsh Shrivastava - 6 years, 2 months ago

" even I had to bash this question!"

What did you bashed??

Harsh Shrivastava - 6 years, 2 months ago

Keshav Bassi
Aug 20, 2015

The first three digit number whose sum of digits is 5 is 104. The next nuber is 113. We will find that difference between all these numbers comes to be 9 in each case. Using AP we can find the answer i.e. 900/9=100

Some Sum Semantics

The answer is 100.

2 solutions

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