Factors, Primes & Composite Numbers
... If one of the factors is divisible by 2, write it down and find the next factor. If not, check if the factor is evenly divisible by 3, 5, 7, 9, etc. ...
... If one of the factors is divisible by 2, write it down and find the next factor. If not, check if the factor is evenly divisible by 3, 5, 7, 9, etc. ...
Revisiting a Number-Theoretic Puzzle: The Census
... Table 1: Triples whose product is 36. The last question of the census taker is important. Its answer decides which of the triples {9, 2, 2} and {6, 6, 1} gives the ages of the daughters. What if the mother’s answer to the last question is negative? Does it point to the triple {6, 6, 1}? Not necessar ...
... Table 1: Triples whose product is 36. The last question of the census taker is important. Its answer decides which of the triples {9, 2, 2} and {6, 6, 1} gives the ages of the daughters. What if the mother’s answer to the last question is negative? Does it point to the triple {6, 6, 1}? Not necessar ...
Algorithms with numbers
... {0, 1, . . . , N − 1} and wraps around whenever you try to leave this range—like the hand of a clock (Figure 1.3). Another interpretation is that modular arithmetic deals with all the integers, but divides them into N equivalence classes, each of the form {i + kN : k ∈ Z} for some i between 0 and ...
... {0, 1, . . . , N − 1} and wraps around whenever you try to leave this range—like the hand of a clock (Figure 1.3). Another interpretation is that modular arithmetic deals with all the integers, but divides them into N equivalence classes, each of the form {i + kN : k ∈ Z} for some i between 0 and ...
Inclusion-Exclusion Principle
... multiples of 2 and 3 for instance. We must subtract 16 multiples of 6, 10 multiples of 10 and 6 multiples of 15. It seems as if 50 + 33 + 20 − 16 − 10 − 6 = 71 is the final answer, but it is not! The multiples of 30 were counted 3 times and eliminated 3 times. They are not accounted for. We have to ...
... multiples of 2 and 3 for instance. We must subtract 16 multiples of 6, 10 multiples of 10 and 6 multiples of 15. It seems as if 50 + 33 + 20 − 16 − 10 − 6 = 71 is the final answer, but it is not! The multiples of 30 were counted 3 times and eliminated 3 times. They are not accounted for. We have to ...
Sample pages 1 PDF
... functions: one can really compute their value at a complex number α. This problem explains a highfalutin’ way of interpreting rational numbers as functions too. i ) First of all, show that we can identify the set of complex numbers α with the set of maximal ideals in C[X] via the correspondence α ↔ ...
... functions: one can really compute their value at a complex number α. This problem explains a highfalutin’ way of interpreting rational numbers as functions too. i ) First of all, show that we can identify the set of complex numbers α with the set of maximal ideals in C[X] via the correspondence α ↔ ...
Fermat`s Little Theorem and Chinese Remainder Theorem Solutions
... Thus d|p. Observe that d 6= 1, and so d = p. Now by Fermat’s Little Theorem, 2q−1 ≡ 1 (mod q), so d = p divides q − 1. This implies that p ≤ q − 1, so q > p. A consequence of this result is the fact that there are infinitely many prime numbers. This was known by the mathematicians of ancient Greece. ...
... Thus d|p. Observe that d 6= 1, and so d = p. Now by Fermat’s Little Theorem, 2q−1 ≡ 1 (mod q), so d = p divides q − 1. This implies that p ≤ q − 1, so q > p. A consequence of this result is the fact that there are infinitely many prime numbers. This was known by the mathematicians of ancient Greece. ...
Modular Arithmetic - Jean Mark Gawron
... prime to 26; 6 is not. 26 and 6 are both even and thus share the factor 2. All even numbers share a factor of 2 with 26. All have multiplication tables that begin repeating at 13 (check this out). 13 evenly divides 26. Thus it shares the common factor 13 with 26; its division table contains even mor ...
... prime to 26; 6 is not. 26 and 6 are both even and thus share the factor 2. All even numbers share a factor of 2 with 26. All have multiplication tables that begin repeating at 13 (check this out). 13 evenly divides 26. Thus it shares the common factor 13 with 26; its division table contains even mor ...
