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# System prepended metadata

title: Prove that the sum of two even

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3. Prove that the sum of two even numbers n1 and n2 is even.
let both n1 and n2 be even. Then by definition there ixist integers K1 and K2
n1 = 2k1
n2 = 2k2
The sum of n1 + n2 = 2k1 + 2k2 which turns into 2(k1+k2) by distributive     property
Now we have the definition of an even number which is an integer (k1+k2) that is multiplied by 2.

9. Scary Clown offers a Sad Meal containing a sandwich, salad, a dessert, and a drink. (They are not mixed together in the box). There are 11 types of sandwiches, 3 types of salads, and 5 different kinds of desserts. A person with low standards for food could eat a different Sad Meal every day for three years. So how many drinks are possible choices for a Sad Meal?
So first we calculate the amount of options possible for the meals with the product principle
11 Sandwiches
3 salads
5 deserts
11 x 3 x 5 = 165
since there are 165 different combinations of meals that means that there are also the same mount of diferent drinks. 165
20. How many different seven-digit phone numbers begin with 231- and contain no 9s?
There are 9 different numbers for each of the four avaiable digits. So we would do 9 to the 4th power 9 x 9 x 9 x 9 which is 6561.
24. . A cold-footed centipede has a drawer filled with many, many socks. And yes, the centipede does have 100 feet. If the centipede only owns green and brown socks, how many must it pull from the drawer in the dark of the morning to be sure it has a matching set for all of its feet (100 socks of the same color)? What if the centipede also owns polkadotted socks? What if the centipede’s drawer has many, many socks of k different colors?
First it would need to pull at least 199 different socks to ensure the same amount of color. And if he has polkadotted socks it would then turn into 199 + polkadots(p) and if he has k different colors it would then turn into 199 + p + k.
30. Magic Trick! You challenge a friend to choose seven different natural numbers in the range 1-12. Prove that two of your friend’s seven numbers sum to 13.
We would use the pigeonhole principle the seven numbers they chose would be pigeons and the amount of numbers would be pigeonholes 12.
Since 2 of the friends are chosing the numbers makes it so that they have 2 x 7 = 14 total pigeons whilst there are only 12 available pigeonholes. And according to the pigeonhole theory if there are more pigeons that pigeonholes one of them will have an extra.