Math for The Modern World

FInd the first 15 pairs of twin primes. A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two.

(3,5)(5,7)(11,13)(17,19)(29,31)(41,43)(59,61)(71,73)(101,103)(107,103)(107,109)(137,139)(149,151)(179,181)(191,193)(197,199)

#27 pg 82

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Fibonacci Bunny Problem

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Solve the following equations for x. x=2+1/x, x=3+1/x

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Where would 129 be on a mod 13 clock (which goes from 0 to 12). Where is the remainder when 129 is divided by 3?

129 = (9 x 13) +12 Spin the clock around 9 times then move the clock hand 12 hours more.

Express the first 15 even numbers greater than 2 as the sum of two prime numbers. Remember: Goldbach's conjecture - Every even integer greater than 2 can be expressed as the sum of two primes.

2 + 2 = 4 3 +3 = 6 3 + 5 = 8 5 + 5 =10 7 + 5 = 12 7 + 7 = 14 5 + 11 = 16 7 + 11 = 18 7 + 13 = 20 5 + 17 = 22 11 + 13 = 24 13 + 13 = 26 11 + 17 = 28 13 + 17 = 30 13 + 19 = 32 17 + 17 = 34

List the first 15 prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Name a perfect number that comes after 6. Perfect Number definition: a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3.

28. 1, 2, 4, 7, 14 = 28

Express each of the following as products of primes. 6, 24, 27, 35,120.

6= 2 x 3 24: 2 x 2 x 2 x 8 27= 3 x 3 x 3 35= 5 x 7 120=2 x 2 x 2 x 3 x 5

Where would 2015 be on a mod 7 clock? What is the remainder when 2015 is divided by 7?

7 x 287 Remainder: 6

Solve for x: x= 1 + 6/x Remember: a=a/1

Expand the fraction. 1/1 + 6/x Multiply. x+1/x+1 +6/x The answer: x+6/x

√3 + √5

No.

We know √2 is irrational. Therefore 3√2 / 5√2 must also be irrational. Is this conclusion correct? Why or why not?

No. √2 cancels to give 3/5.

Can you figure out the next few terms and describe how to find all the terms in the sequence? 1 11 21 1211 111221 312211

Say the sequence. 13112221 1113213211 ...

what proportion of the first 1000 natural numbers have a 3 somewhere in them? for example 135, 403, 339 all contain a three, whereas 402,677 and 8 do not.

There are 10 numbers in 30-39. There are also another 9 numbers in 3,13,23,43,53,63,73,83,93. So that makes 19 in 1-100. There is the same amount in 101-200, 201-300, 401-400, ... , 901-1000. There are 9 of those blocks of numbers. 301-400 have at least 100 numbers containing a three. All together, N= 100 +9x19=100+171=271 R= 271/1000

Is 4/9 rational? Is 1.75 rational? Is √20/3√5 rational?

Yes. Yes. No. Yes

Fibonacci Series:

a series of numbers in which each number ( Fibonacci number ) is the sum of the two preceding numbers. The simplest is the series 1, 1, 2, 3, 5, 8, etc.

Toda is Saturday. What day of the week will it be in 3724 days? What day of the week will it be in 365 days?

is 3724 divisible by 7? Yes. 3724 = 532 x 7 Therefore: if today is saturday it will be saturday in 3724 days. 365 364+1=365 52 x 7 +1 365 = Sunday

If n is an odd number greater than or equal to 3, can n+1 ever be prime? What if n equals 1?

under those conditions n+1 can never be prime because n has to be an odd number and any odd number plus an odd number will give you an even number. And since every even number can be divided by 2 no even number is prime. However if n=1 then it would be prime.