3 Congruence

... Proof: This is little more than a divisibility theorem. Since n|(b − a), we have cn|c(b − a) or cn|(cb − ca), and this is the result. The converse is also valid. Thus, if ca ≡ cb mod cn with c > 0 then a ≡ b mod n. These results can be stated: A congruence can by multiplied through (including the mo ...

... Proof: This is little more than a divisibility theorem. Since n|(b − a), we have cn|c(b − a) or cn|(cb − ca), and this is the result. The converse is also valid. Thus, if ca ≡ cb mod cn with c > 0 then a ≡ b mod n. These results can be stated: A congruence can by multiplied through (including the mo ...

Number Theory & RSA

... Carmichael number: a composite positive integer n which satisfies the equation bn-1 = 1 mod n for all positive integers b which are relatively prime to n Korselt Theorem (1899): A positive odd composite integer n is a Carmichael number if and only if n is square-free, and for all prime divisors p of ...

... Carmichael number: a composite positive integer n which satisfies the equation bn-1 = 1 mod n for all positive integers b which are relatively prime to n Korselt Theorem (1899): A positive odd composite integer n is a Carmichael number if and only if n is square-free, and for all prime divisors p of ...

Elementary Problems and Solutions

... To:Fibonacci product inequality, 6.2(1968)185 So: Untitled 7.1(1969)111 To:No perfect Fibonacci or Lucas numbers, 6.2(1968)186 So: Untitled 7.1(1969)112 To:Special product sequence, 6.4(1968)288 So: A Multiplicative Analogue, 7.2(1969)220 To:Determinant of a Fibonacci-Lucas matrix, 6.4(1968)288 So: ...

... To:Fibonacci product inequality, 6.2(1968)185 So: Untitled 7.1(1969)111 To:No perfect Fibonacci or Lucas numbers, 6.2(1968)186 So: Untitled 7.1(1969)112 To:Special product sequence, 6.4(1968)288 So: A Multiplicative Analogue, 7.2(1969)220 To:Determinant of a Fibonacci-Lucas matrix, 6.4(1968)288 So: ...

Elementary Real Analysis - ClassicalRealAnalysis.info

... We apply these markings to some entire chapters as well as to some sections within chapters and even to certain exercises. We do not view these markings as absolute. They can simply be interpreted in the following ways. Any unmarked material will not depend, in any substantial way, on earlier marked ...

... We apply these markings to some entire chapters as well as to some sections within chapters and even to certain exercises. We do not view these markings as absolute. They can simply be interpreted in the following ways. Any unmarked material will not depend, in any substantial way, on earlier marked ...

... of those with a ﬁxed number of blocks, of those with a given block structure, results on the number of (multi-)chains of a given length in a given poset of (generalised) non-crossing partitions, results on rank-selected chain enumeration (that is, results on the number of chains in which the ranks o ...

College Algebra Week 2

... 1. Remove parentheses and combine like terms on each side 2. Use addition property to get like terms on the same side as each other (things with x on one side, numbers only on the other) 3. Use the multiplication property to get the x alone 4. If x is negative, use the –1 multiplication trick to mak ...

... 1. Remove parentheses and combine like terms on each side 2. Use addition property to get like terms on the same side as each other (things with x on one side, numbers only on the other) 3. Use the multiplication property to get the x alone 4. If x is negative, use the –1 multiplication trick to mak ...

Residue Number systems - IEEE

... CRT: RNS {m1,m2,m3} Residues {x1,x2,x3} Define Mi=M/mi and M=m1m2m3 Decoded Binary number X = [M1{(1/M1) mod m1}x1+ {M2 (1/M2) mod m2}x2+ M3{(1/M3) mod m3}x3]mod M e.g. {3,5,7} M=105, M1=35,M2=21,M3=15 (1/35) mod 3 = 2, (1/21) mod 5=1, (1/15) mod 7=1. X= [70x1+21x2+15x3] mod 105 Consider (1,2,3), X ...

... CRT: RNS {m1,m2,m3} Residues {x1,x2,x3} Define Mi=M/mi and M=m1m2m3 Decoded Binary number X = [M1{(1/M1) mod m1}x1+ {M2 (1/M2) mod m2}x2+ M3{(1/M3) mod m3}x3]mod M e.g. {3,5,7} M=105, M1=35,M2=21,M3=15 (1/35) mod 3 = 2, (1/21) mod 5=1, (1/15) mod 7=1. X= [70x1+21x2+15x3] mod 105 Consider (1,2,3), X ...

Summary of lectures.

... If n is prime this is simply If ax ≡ ay mod p and a 6≡ 0 mod p then x ≡ y mod p. Note the similarity with the familiar algebraic law if congruence is replace by equality: If ax = ay and a 6= 0 then x = y. The result for primes is equivalent to If ab ≡ 0 mod p then a ≡ 0 mod p or b ≡ 0 mod p. Using t ...

... If n is prime this is simply If ax ≡ ay mod p and a 6≡ 0 mod p then x ≡ y mod p. Note the similarity with the familiar algebraic law if congruence is replace by equality: If ax = ay and a 6= 0 then x = y. The result for primes is equivalent to If ab ≡ 0 mod p then a ≡ 0 mod p or b ≡ 0 mod p. Using t ...

Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. The most basic topics in elementary mathematics are arithmetic and geometry. Beginning in the last decades of the 20th century, there has been an increased emphasis on problem solving. Elementary mathematics is used in everyday life in such activities as making change, cooking, buying and selling stock, and gambling. It is also an essential first step on the path to understanding science.In secondary school, the main topics in elementary mathematics are algebra and trigonometry. Calculus, even though it is often taught to advanced secondary school students, is usually considered college level mathematics.