Memorable Year

For students of class 6-8 (age 12 to 14)
User avatar
sakib17442
Posts:15
Joined:Sun Apr 11, 2021 10:04 pm
Contact:
Memorable Year

Unread post by sakib17442 » Mon Jul 12, 2021 12:44 am

Sudipta noticed that his birth year $1978$
is a memorable year in which two number of two sides are $19$ and $78$. The sum of this two number is $97$ which is it's middle number. Now it is $2018$. How many years will Sudipta wait for the next memorable year?

Source : Gonitzoggo, Link: https://gonitzoggo.com/archive/problem/178
Games You can't win because you'll play against yourself.
---Dr.Seuss

User avatar
Anindya Biswas
Posts:264
Joined:Fri Oct 02, 2020 8:51 pm
Location:Magura, Bangladesh
Contact:

Re: Memorable Year

Unread post by Anindya Biswas » Sat Jul 17, 2021 4:11 pm

sakib17442 wrote:
Mon Jul 12, 2021 12:44 am
Sudipta noticed that his birth year $1978$
is a memorable year in which two number of two sides are $19$ and $78$. The sum of this two number is $97$ which is it's middle number. Now it is $2018$. How many years will Sudipta wait for the next memorable year?

Source : Gonitzoggo, Link: https://gonitzoggo.com/archive/problem/178
Let's assume $1000a+100b+10c+d$ is a memorable year where $a,b,c,d\in\mathbb{Z}; 2\leq a\leq9; 0\leq b,c,d\leq9$. [That means they are valid digits in base $10$.]

They must satisfy the equation,
$10a+b+10c+d=10b+c$
$\Longleftrightarrow 10a+9c+d=9b$
To get the minimum such year after $2018$, let's guess $a=2$.
So, $20+d=9(b-c)$.
Since $9\mid 20+d$ and $20\leq 20+d\leq 29$,
We must have, $20+d=27\Longleftrightarrow d=7$

From here, we conclude, $b-c=3$. To minimize the value of $b$, let's use the solution $b=3, c=0$.
From here, we conclude that $2307$ is the closest memorable year after $2018$.

So, Sudipta should wait about $\boxed{2307-2018=289}$ years only.
Hoping that people remember this post after this many years and celebrates with great joy :P
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Post Reply