Chapter 6: Mathematics Flashcards
What is number theory?
number theory is the study of the natural numbers 1,2,3
What is prime factorization?
breaking down a large number into the prime numbers that times together to make that number
What is a prime number?
a number that can only be divided equally by itself and 1, ie 11. 1x11 is the only way to get 11 by times
what is a factor?
a factor is any number that can divide into another number evenly (without leaving a remainder) i.e. 12 = 3x4 3 & 4 are factors, but they are not prime factors because 4 is not prime
what is a prime factor?
when all the numbers that can divide into another number are prime numbers, 12 = 3x2x2 all prime numbers
what is the primer factorization of 8?
2x2x2
what is the prime factorization of 30
2x3x5
what is the prime factorization of 15
3x5
What does LCM stand for
Least common mulitple (some places say lowest common multiple)
What would be multiples of 3 & 5?
15, 30, 45, 60 etc
what would be the Least Common Multiple of 3 & 5?
15
Using Least Common Multiple helps you to find what fractions item easier?
LCM helps you to quickly find the common denominator to add or subtract fractions
How can you quickly find the LCM of two numbers?
do the prime factorization of each number, see how many times each prime number appears and record the most times, then times them all together
What is the prime factorization way of finding the LCM of 6 and 28?
6=2x3
28=2x2x7
so 2x2x3x7 = 84
so the LCM of 6 & 28 is 84
What is the prime factorization of 18?
2x3x3
What is the prime factorization of 75?
2x5x5
What is the prime factorization of 18 & 75?
2x3x3=18
2x5x5=75
3x3 x2x5x5=450
What is Guass Summation?
Pairing up numbers so they equal the same amount when you are adding a sequence, thus saving you time
What would be the sum of 1-30 if you add each number consecutively? Do the pairs trick from page 90
1-10 = 1+10=11 for each of the 5 pairs 55
11-20 = 11+20=31 for each of the 5 pairs 155
21-30 = 21+30=51 for each of the 5 pairs 255
55+155+255=465
Did you notice the pattern in the way the numbers added up in each section?
What is modular arithmetic?
clock math
what is clock math?
we have a special way of counting our hours, so that when we get past noon or midnight we go back to 1
If you are leaving home at 10 am and travel for 6 hours, what time will you get to your destination?
4pm
BUT did you just add 10+6? shouldn’t we say 16? In the 24 hour clock we would say 16, so in modular arithmetic or clock math 4=16 (just think 24 hour clock time to figure it out)
What is Congruent Modulo
Congruent (same) modulo is when numbers have the same difference between them always. i.e. 12,24,36 are congruent modulo 12
How would we show that 2 and 14 are the same on a clock?
2=14(mod 12)
what is the mathematical symbol for congruency?
=
Why is 3=39(mod12) ? why isn’t it mod 36?
36 is divisible by 12, which is why they put mod12
If there was a 7 hour clock instead of 12. You would say two numbers are modulo 7 if their difference was divisible by 7
2=9(mod7) is correct
In a 7 hour clock world, how would you show 5 and 14?
5=14(mod7)
Is 3=9(mod7) correct?
no 9-3=6 so not divisible by 7
Is 1=7(mod7) correct?
no 7-1=6 so not divisible by 7
is 2=16(mod7) correct?
yes 16-2=14 which is divisible by 7
What is modulo arithmetic?
using the mod, such as the 12 hour clock, to show the equivalent place not the clock in 24 or 12 hour time. i.e. 2(mod12) is the same as 14
What’s a quick way to figure out a mod answer?
use division 9+5=14, mod 12 so divide 14 by 12 = 1 remainder 2 answer is 2(mod12) weirdly the number of times it divides is not important in modulo arithmetic
Use mod 7 and figure out 3+18
3+18=21
21 divided by 7 = 3 no remainder
ANSWER: 0(mod7)
use mod 7 to figure out 2+4
2+4=6
6 divided by 7 = 0 with a remainder of 6
ANSWER: 6(mod7)
use mod 7 to figure out 19+10
19=10=29
29 divided by 7 = 4 remainder 1
1(mod7)
could you do math in modulo 890009?
yes the number doesn’t matter, it is just showing the consistent number that is between each set, i.e.: 1, 890010 is mod 890009
