Elementary Number Theory

Follow the same saving procedures as in the previous worksheet.

We continue by experimenting with some more very basic calculations. Again the intent is to learn some basic MAPLE syntax, while observing more neat features of a Computer Algebra System (CAS). We will also review some basic concepts from number theory.

perform the following calculations with MAPLE. As you do these exercises enter notes and comments onto your worksheet in text mode. As always questions/comments in bold face require written response.

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

1. What is gcd an abbreviation for?

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

2. What is lcm an abbreviation for?

3. Compute gcd(6,8)lcm(6,8).

Execute the following command sequence:

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

%*%%;

140*958;

4. What general formula is suggested by these calculations? Test the formula with several more pairs of integers.

Execute the following:.

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

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYoLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GKC8lJ29wYXF1ZUdGOC8lK2V4ZWN1dGFibGVHRjgvJSlyZWFkb25seUdGOC8lKWNvbXBvc2VkR0Y4LyUqY29udmVydGVkR0Y4LyUraW1zZWxlY3RlZEdGOC8lLHBsYWNlaG9sZGVyR0Y4LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZHLyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1JJm1mcmFjR0YlNiotRiQ2JUYsLUYkNiYtRi02OVEkbGNtRihGMEYzRjZGOUY8Rj5GQEZCRkVGSC9GS0Y7RkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8tSSNtb0dGJTYzUTAmQXBwbHlGdW5jdGlvbjtGKC8lJWZvcm1HUSZpbmZpeEYoLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSdsc3BhY2VHUSQwZW1GKC8lJ3JzcGFjZUdGZnAvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKG1heHNpemVHUSlpbmZpbml0eUYoLyUobWluc2l6ZUdRIjFGKC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUwZm9udF9zdHlsZV9uYW1lR0ZYLyUlc2l6ZUdGNS8lK2ZvcmVncm91bmRHRkQvJStiYWNrZ3JvdW5kR0ZHLUYkNiUtRmpvNjNRIihGKC9GXnBRJ3ByZWZpeEYoL0ZhcEY7RmJwL0ZlcFEudGhpbm1hdGhzcGFjZUYoL0ZocEZqci9GanBGO0ZbcUZdcUZgcUZjcUZlcUZncUZpcUZbckZdckZfci1GJDYlLUkjbW5HRiU2OVEjMzhGKEYwRjNGNi9GOkY4RjxGPkZARkJGRUZIRmhvRkxGTkZQRlJGVEZWRllGZW5GZ24vRmpuUSdub3JtYWxGKEZcby1Gam82M1EiLEYoRl1wRmBwL0ZjcEY7RmRwL0ZocFEzdmVyeXRoaWNrbWF0aHNwYWNlRihGaXBGW3FGXXFGYHFGY3FGZXFGZ3FGaXFGW3JGXXJGX3ItRmBzNjlRJDEwMEYoRjBGM0Y2RmNzRjxGPkZARkJGRUZIRmhvRkxGTkZQRlJGVEZWRllGZW5GZ25GZHNGXG8tRmpvNjNRIilGKC9GXnBRKHBvc3RmaXhGKEZockZicEZpci9GaHBRMnZlcnl0aGlubWF0aHNwYWNlRihGXHNGW3FGXXFGYHFGY3FGZXFGZ3FGaXFGW3JGXXJGX3JGLEYsLUYkNiVGLC1GJDYmLUYtNjlRJGdjZEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGaG9GTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcb0Zpb0ZhckYsRiwvJS5saW5ldGhpY2tuZXNzR1EiMUYoLyUrZGVub21hbGlnbkdRJ2NlbnRlckYoLyUpbnVtYWxpZ25HRmJ1LyUpYmV2ZWxsZWRHRjhGXXJGX3JGLDcjNiMqJi0lJGxjbUc2JCIjUSIkKyIiIiItJSRnY2RHRlx2ISIi

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

Do bigger integers necessarily yield bigger lcm's? Explain.

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

Two integers are "relatively prime" if the biggest integer that divides both of them is 1.

5. List all the pairs of integers from 100 to 110 that are relatively prime. (Here you might want to look at the MAPLE Programming Supplement to learn how to do "for" and "if" programs.)

An integer is called "prime" if its only divisors are 1 and itself.

The MAPLE command "isprime" determines whether or not a given integer is prime.

6. Use isprime it to see if the following numbers are prime: 29, 35, 1537, your ss#, 10!+1, 329891.

7. List the prime numbers between 1 and 100.

One way to get a list of the primes.

The FUNDAMENTAL THEOREM OF ARITHMETIC states that every integer can be factored in a unique way into a product of powers of primes.

8. Use paper and pencil to factor the following into their prime decompositions: 10! , 340704.

ANSWERS:

The MAPLE command "ifactor" does this automatically.

9. Use ifactor to verify your results from above. Now factor the following numbers: 10!+1 , your ss#.

Execute the following command sequence:

2744280=ifactor(2744280); 267300=ifactor(267300);

2744280*267300=ifactor(2744280*267300);

gcd(2744280,267300)=ifactor(gcd(2744280,267300));

lcm(2744280,267300)=ifactor(lcm(2744280,267300));

lcm(2744280,267300)*gcd(2744280,267300)=ifactor(lcm(2744280,267300)*gcd(2744280,267300));

10. Using the FUNDAMENTAL THEOREM OF ARITHMETIC, provide a written argument to verify the formula gcd(m,n)lcm(m,n) = mn. ARGUMENT:

Demonstrate your argument with two sufficiently interesting integers (different from above) and the ifactor command in MAPLE.

For two integers m & n with 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 we say the "remainder of n divided by m" is r if

n = qm + r, for integers q >0 & r with 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 < m.

11. With paper and pencil compute the following remainders and include your answers: 24 divided by 5, 156 divided by 7, 32 divided by 2, 57 divided by 4. ANSWERS:

In MAPLE this remainder is denoted by "n mod m." Verify the above answers with MAPLE.

12. Compute the following remainders:

400*23 mod 19;

1321 mod (5*7);

13. Explain why an integer n is "even" exactly when n mod 2 = 0, and is "odd" exactly when

n mod 2 = 1. EXPLANATION

14. Explain why every product of two odd integers must be an odd integer and every product of an even integer with another integer must be even. EXPLANATION:

15. Do the following calculations in your head and record the answers.

ANSWERS:

Check the results with MAPLE.