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# 4383_Notes_4o3_fill - Binary can be useful calculating...

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Math 4383 Section 4.3 Page 1 of 6 Number Theory Chapter 4 Section 4.3 – Binary and Decimal Representation of Integers Place Value Systems of Numeration Expanded form in base “b” Short form –

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Math 4383 Section 4.3 Page 2 of 6 We work in base 10, WHY? 3526 – The base 10 place value system gives us some hints on congruence easily A and B are congruent mod 10 if they have the same last digit,… Congruent mod 100 if …. Also, consider the classic test for divisibility by 9 – How do you tell if a 3 digit number is divisible by 11?
Math 4383 Section 4.3 Page 3 of 6 Another base that is very useful to number theory is base 2, also called binary Digits are 0 and 1 – place values are powers of 2

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Unformatted text preview: Binary can be useful calculating powers mod n Math 4383 Section 4.3 Page 4 of 6 Let’s revisit the divisibility test for 9 and 11 in a little more sophisticated way Theorem: If ( 29 mod a b n ≡ and P(x) is any polynomial with integer coefficients, then ( 29 ( 29 ( 29 mod P a P b ≡ . Proof: Math 4383 Section 4.3 Page 5 of 6 Consider a place value number as a polynomial with the base “b” the value of x – So, the base ten number 35712 can be thought of as ( 29 4 3 2 3 5 7 1 2 P x x = + + + + evaluated when x = 10. Why is a number divisible by 9 when the sum of its digit is? Math 4383 Section 4.3 Page 6 of 6 What about divisibility by 11?...
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4383_Notes_4o3_fill - Binary can be useful calculating...

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