Wikipedia
Number theory - Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into several fields, according to the methods used and the type..
Number theory - Number theory is the branch of pure mathematics concerned with the properties of number s in general, and integer s in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into several fields, according to the methods used and..
Encyclopedia
number theory - number theory branch of mathematics concerned with the properties of the integers (the numbers 0, 1, -1, 2, -2, 3, -3, …). An important area in number theory is the analysis..
number theory - number theory branch of mathematics concerned with the properties of the integers (the numbers 0, 1, -1, 2, -2, 3, -3, …). An important area in number theory is the analysis of prime numbers. A prime number is an integer p >1 divisible only by 1 and p ; the first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. Integers that have other divisors are called composite; examples are 4, 6, 8, 9, 10, 12, … . The fundamental theorem of arithmetic, the unique factorization theorem, asserts that any positive integer a is a product ( a  =  p1  ·  p2  ·  p3  · · ·  pn ) of primes that are unique except for the order in which they are listed; e.g., the number 20 is the product 20 = 2 · 2 ·5, and it is unique (disregarding order) since 20 has this and only this product of primes. This theorem was ....
TutorVista
Rational and Irrational Numbers
. Problems involving rational numbers are simplified using 'BODMAS' rul..
Questions on Number theory
Question 1 - Question: Define an even number. Answer: A number which is divisible by 2 is called an even number..
Basic number theory question and answers
Question 1 - Question: List the elements of the following sets. Answer: i) Omit o and u from the vowels a, e, i, o, u to get {a,e,i} ii) There are exactly two such names {Karnataka, Keral..
Question 1 - Question: List the elements of the following sets. Answer: i) Omit o and u from the vowels a, e, i, o, u to get {a,e,i} ii) There are exactly two such names {Karnataka, Keral.. Math Problem Solved #2 (Number Theory)The following problem came from a friend in need of math help. Disprove the statement: There exists an integer, "n" such that n^3 - n + 1 is even. The preceding was disproved for ALL "n" by showing that no matter what number is used for "n", the result will always be odd.
please give me links about number theory problems with answers.?to Ron Paul: 'Now, I've met a lot of your supporters online, but I've noticed that a good number of them seem to buy into this conspiracy theory regarding the Council on Foreign Relations and some plan to make a North American Union by merging the United States with Canada and Mexico. These supporters of yours seem to think that you also believe in this theory. 'So my question to you is, do you really believe in all this, or are people just putting words in your mouth?' Answer: 'Well, that ...
Question : For each positive integer k, let f(k)=9k+7. How many perfect squares are there among the integers f(1), f(2), ...,f(1000)?
Answer : MathJakk says that: f(1) = 16 = 4 f(1000) = 9007 = 94 That's actually wrong. 94 = 8836 That's MathJakk's other mistake. He seems to have counted all the squares between f(1) and f(1000), but not all of them are represented as f(k) for some integer. 8836 is not - no number k will give you f(k) = 8836. The question actually is: between f(1) = 16 and f(1000) = 9007, how many squares are there that are congruent to 7, modulo 9? That's the real question. 4 is a start, since it's the first. 4 = 16 7 (mod 9) 5 = 25 7 (mod 9) 6 = 36 0 (mod 9) 7 = 25 4 (mod 9) 8 = 64 1 (mod 9) 9 = 81 0 (mod 9) 10 = 100 1 (mod 9) 11 = 121 4 (mod 9) 12 = 144 0 (mod 9) From there, it repeats. Why? Because the numbers start repeating, mod 9, so their squares must repeat as well. So the pattern is: 7 7 0 4 1 0 1 4 0 How many times do we go through that pattern? Well, our illustriously incorrect friend MathJakk (who has now changed his answer ..
Answer : MathJakk says that: f(1) = 16 = 4 f(1000) = 9007 = 94 That's actually wrong. 94 = 8836 That's MathJakk's other mistake. He seems to have counted all the squares between f(1) and f(1000), but not all of them are represented as f(k) for some integer. 8836 is not - no number k will give you f(k) = 8836. The question actually is: between f(1) = 16 and f(1000) = 9007, how many squares are there that are congruent to 7, modulo 9? That's the real question. 4 is a start, since it's the first. 4 = 16 7 (mod 9) 5 = 25 7 (mod 9) 6 = 36 0 (mod 9) 7 = 25 4 (mod 9) 8 = 64 1 (mod 9) 9 = 81 0 (mod 9) 10 = 100 1 (mod 9) 11 = 121 4 (mod 9) 12 = 144 0 (mod 9) From there, it repeats. Why? Because the numbers start repeating, mod 9, so their squares must repeat as well. So the pattern is: 7 7 0 4 1 0 1 4 0 How many times do we go through that pattern? Well, our illustriously incorrect friend MathJakk (who has now changed his answer ..
Question : Could anyone help with this:
Find the smallest positive integers that has exactly 30 positive integer divisors.
Answer : Let n = p1^a1 * ... * pr^ar where p1, p2, ... are distinct primes. Then, the number of divisors of n is (a1+1) * ... (ar +1). Since 30 = (i) 2 * 15, (ii) 3 * 10, (iii) 5 * 6, or (iv) 2 * 3 * 5, we have four cases: (i) exponents are 1 and 14: (2^14) * (3^1) = 49152 is the smallest such integer. (ii) exponents are 2 and 9: (2^9) * (3^2) = 4608 is the smallest such integer. (iii) exponents are 4 and 5: (2^5) * (3^4) = 2592 is the smallest such integer. (iv) exponents are 1, 2, and 4: (2^4) * (3^2) * (5^1) = 720 is the smallest such integer. From these four cases, we see that 720 is the smallest integer with exactly 30 positive divisors.
Answer : Let n = p1^a1 * ... * pr^ar where p1, p2, ... are distinct primes. Then, the number of divisors of n is (a1+1) * ... (ar +1). Since 30 = (i) 2 * 15, (ii) 3 * 10, (iii) 5 * 6, or (iv) 2 * 3 * 5, we have four cases: (i) exponents are 1 and 14: (2^14) * (3^1) = 49152 is the smallest such integer. (ii) exponents are 2 and 9: (2^9) * (3^2) = 4608 is the smallest such integer. (iii) exponents are 4 and 5: (2^5) * (3^4) = 2592 is the smallest such integer. (iv) exponents are 1, 2, and 4: (2^4) * (3^2) * (5^1) = 720 is the smallest such integer. From these four cases, we see that 720 is the smallest integer with exactly 30 positive divisors.