# Maths Primes and Multiples Investigation

Extracts from this document...

Introduction

1. A)I) ?(3)-1, 2=2 II) ?(8)-1, 2, 3, 4, 5, 6, 7=4 III) ?(11)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10=10 IV) ?(24)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23=8 B)I) ?(5)-1, 2, 3, 4=4 II) ?(10)- 1, 2, 3, 4, 5, 6, 7, 8, 9=4 III) ?(15)- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14=8 IV) ?(20)- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19=8 V) ?(50)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49=20 2. A)I) ?(7x4) = ?(7) x ?(4) 7x4=28 ?(28)= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27=12 ?(7)=1, 2, 3, 4, 5, 6=6 ?(4)=1, 2, 3=2 2x6=12 12=12, therefore a prime and an even work(non-prime). B) ?(6x4) = ?(6) x ?(4) 6x4=24 ?(24)=8 ?(6)=1, 2, 3, 4, 5=2 ?(4)=2 2x2=4 4=8, therefore two evens don't work. C) ?(5x10)= ?(50) ?(50)=20 ?(10) x ?(5) 4x4=16 16=20, therefore two multiples don't work. ?(13x3)= ?(39) ?(39)= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38=24 ?(13)=12 ?(3)=2 2x12=24 24=24, therefore two primes work. ...read more.

Middle

The only numbers that don't work all the time are two evens. Now I will try a different format of table to see if it shows me any more correlations. 1st number 2nd Number Works? Prime Non-prime Yes Prime Prime Yes Prime Non-prime Yes Prime Non-prime No Non-prime Non-prime No Prime Non-prime No Prime Non-prime No The only two numbers that work all the time are two primes. The two numbers that always work are two primes. They have no multiples The two numbers that never work are too evens. All evens have the at least on multiple, because two goes into all evens. Therefore two evens all have the same lowest common factor. I will now look at some of the numbers that sometimes work to see why they only work on some occasions. I will use the example of two non-primes: 4 and 6 doesn't work. 4 and 9 does work. Four and six are both evens and therefore have the same common factor. 4 and 9 don't have the same lowest common factor, that's why evens and odd never work. I will check this with the second example of an even and a non-prime: 4 and 8 doesn't work 8 and 9 does work. Four and eight are again both evens and therefore have the same common factor of two. Eight and nine don't have the same lowest common factor, and they therefore don't work, odds and even never work. From these two examples I can see that two numbers with the same common factor don't work. I can therefore do an example that I have not done before to test this, ten and eleven. ...read more.

Conclusion

Therefore their multiples will not have a lowest common factor, and they would always work using this formula. For example, if I used 3 and 5, ?(32 x 52) ?(225)=120 ? (9)=6 ? (25)=20 6 x 20=120 120=120, so the powers of two primes work, as they don't have a common factor As I have shown, the powers of primes do not have a common factor. I can therefore put the formula for this section into an easier form: ?(pn) x ?(qm) Having the formula in this form will be easier because I will not have to work out the phis of very large numbers to confirm my findings. I can use some of the data that I found in section C) to help me find the phis of some numbers. I can now start to do some more numbers to try and build up a picture using the formula ?(pn) x ?(qm): ? (33)=18 ? (52)=20 18 x 20=360 ? (34)=54 ? (52)=20 54 x 20=1080 ? (32)=2 ? (53)=100 2 x 100=200 ? (32)=2 ? (54)=500 2 x 500=1000 ? (52)=20 ? (72)=42 20 x 42=840 ? (53)=100 ? (72)=42 100 x 42=42,000 ? (54)=500 ? (72)=42 42 x 500=21,000 ? (52)=20 ? (73)=84 20 x 84=1680 ? (52)=20 ? (74)=2058 20 x 2058=41,160 I will now out all of my numbers in a table to try and compare them: 1st Number Power 2nd Number Power Phi 3 2 5 2 120 3 3 5 2 360 3 4 5 2 1,080 3 2 5 3 200 3 2 5 4 1,000 5 2 7 2 840 5 3 7 2 42,000 5 4 7 2 21,000 5 2 7 3 1680 5 2 7 4 41,160 ?? ?? ?? ?? The Phi Function ...read more.

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