How do I start?

$I\quad came\quad across\quad the\quad following\quad question:\\ Prove\quad that,\quad given\quad any\quad 52\quad integers,\quad there\quad exist\quad two\quad ofthem\quad whose\quad sum,\quad or\quad else,\quad whose\quad difference,\quad is\quad divisible\quad by\quad 100.$

Please help. I don't know how to start the question on the first place.

$I\quad just\quad started\quad using\quad latex.\quad Sorry$

#Algebra

Easy Math Editor

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Comments

$Bad English In the question. Sorry$

Ceesay Muhammed - 6 years, 5 months ago

It's something about the pigeonhole principle. Sorry no time now, got to go to class.

Marc Vince Casimiro - 6 years, 5 months ago

Since we need to check the divisibility by $100$ , we'll only concentrate on the last two digits of the numbers.

The only possible last two digits are $\text{00, 01, 02, ..........., 98, and 99}$ , a total of $100$ last two digits.

If the last two digits of any $2$ numbers are the same, their difference will be divisible by $100$ .

Hence, we know that the last two digits of all numbers are different. Now, we divide the numbers into pairs that add up to $100$ . The pairs are $(01,99)$ , $(02,98)$ , ........., $(49,51)$ . And we can also use $0$ and $50$ only once.

Total number of pairs is $49$ and $2$ extra numbers. This makes a total of $51$ numbers, but we need $52$ .

Hence, we must either use two numbers from any pair (making the sum divisible by $100$ ), or we need to use a number twice (making the diifference divisible by $100$ ).

$\Large \mathbb{QED}$

Satvik Golechha - 6 years, 5 months ago

@Ceesay Muhammed you don't need to latex the whole text; see the Beginner Latex Guide, it'll help a lot.

Satvik Golechha - 6 years, 5 months ago

Q.E.D. is an understatement :D

Marc Vince Casimiro - 6 years, 5 months ago

LOL It's not even a statement... :-P

Satvik Golechha - 6 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$