Integer Definition and 620 Threads

An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold



(

Z

)


{\displaystyle (\mathbb {Z} )}
letter "Z"—standing originally for the German word Zahlen ("numbers").ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.

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  1. bob012345

    I Integer Square Roots of P{0,1,2,4,5,6,7,8,9}

    This is not homework but just a mental exercise. I suspect there are no integer square roots in this set but can't prove it. I mean all digits are used once in each number. I am interested in arguments from principal and not algorithms. Note the number ##3## is missing. Thanks.
  2. al4n

    B How to find an integer solution to a nonlinear equation?

    given something like: an = c where c is given and a, n, and c are only allowed to be integers. how would one find the value of say n or a?
  3. V

    Proving an integer is composite

    I am really struggling in how to begin this problem. So far I have considered using the Euclidean Algorithm and trying to find the gcd of each number like gcd(9,10) but each time they give me 1 so that doesn't work. My next idea is to do a proof by contradiction where I start with assuming that...
  4. M

    Finding Integer Solutions to Polynomial Equations: Can it be Done Easily?

    Hello, Am re-studying math & calculus aiming to start pure math studying later. However, I got this problem in Stewart calculus. Typically, this is a straightforward IVT application. x = x^3 + 1, call f(x)= x^3 - x + 1 & apply IVT. However I have two things to discuss. First thing is simple...
  5. M

    Is the following true, for each positive integer ## k ##?

    Let ## 5k+4=p_{1}^{k_{1}}p_{2}^{k_{2}}\dotsb p_{s}^{k_{s}} ##. Then ## 5\equiv 0\pmod {5} ## and ## 5k+4\equiv 4\pmod {5} ##. Thus ## p_{i}^{k_{i}}\not \equiv 0\pmod {5} ## for ## i=1, 2,..., s ##. Suppose all ## p_{i}^{k_{i}}\equiv 1\pmod {5} ##. Then ## p_{1}^{k_{1}}p_{2}^{k_{2}}\dotsb...
  6. M

    Find an integer having the remainders ## 1, 2, 5, 5 ##

    Let ## x ## be an integer. Then ## x\equiv 1\pmod {2}, x\equiv 2\pmod {3}, x\equiv 5\pmod {6} ## and ## x\equiv 5\pmod {12} ##. Note that ## x\equiv 5\pmod {6}\implies x\equiv 5\pmod {2\cdot 3} ## and ## x\equiv 5\pmod {12}\implies x\equiv 5\pmod {3\cdot 4} ##. Since ## gcd(2, 3)=1 ## and ##...
  7. M

    Find an integer having the remainders ## 2, 3, 4, 5 ##.

    Let ## x ## be an integer. Then ## x\equiv 2\pmod {3}, x\equiv 3\pmod {4}, x\equiv 4\pmod {5} ## and ## x\equiv 5\pmod {6} ##. This means \begin{align*} &x\equiv 2\pmod {3}\implies x+1\equiv 3\pmod {3}\implies x+1\equiv 0\pmod {3},\\ &x\equiv 3\pmod {4}\implies x+1\equiv 4\pmod {4}\implies...
  8. M

    Finding Integer with Chinese Remainder Theorem

    Consider a certain integer between ## 1 ## and ## 1200 ##. Then ## x\equiv 1\pmod {9}, x\equiv 10\pmod {11} ## and ## x\equiv 0\pmod {13} ##. Applying the Chinese Remainder Theorem produces: ## n=9\cdot 11\cdot 13=1287 ##. This means ## N_{1}=\frac{1287}{9}=143, N_{2}=\frac{1287}{11}=117 ## and...
  9. M

    Find the smallest integer ## a>2 ## such that ## 2\mid a ##

    Let ## a>2 ## be the smallest integer. Then \begin{align*} &2\mid a\implies a\equiv 0\pmod {2}\implies a\equiv 2\pmod {2}\\ &3\mid (a+1)\implies a+1\equiv 0\pmod {3}\implies a\equiv -1\pmod {3}\implies a\equiv 2\pmod {3}\\ &4\mid (a+2)\implies a+2\equiv 0\pmod {4}\implies a\equiv -2\pmod...
  10. M

    Determine whether the integer ## 1010908899 ## is divisible by....

    Consider the integer ## 1010908899 ##. Observe that ## 7\cdot 11\cdot 13=1001 ##. Then ## 10^{3}\equiv -1\pmod {1001} ##. Thus \begin{align*} &1010908899\equiv (1\cdot 10^{9}+10\cdot 10^{6}+908\cdot 10^{3}+899)\pmod {1001}\\ &\equiv (-1+10-908+899)\pmod {1001}\\ &\equiv 0\pmod {1001}.\\...
  11. M

    Proof That 6 Divides Any Integer N

    Proof: Suppose that ## 6 ## divides ## N ##. Let ## N=a_{m}10^{m}+\dotsb +a_{2}10^{2}+a_{1}10+a_{0} ##, where ## 0\leq a_{k}\leq 9 ##, be the decimal expansion of a positive integer ## N ##. Note that ## 6=2\dotsb 3 ##. This means ## 2\mid 6 ## and ## 3\mid 6 ##. Then ## 2\mid...
  12. M

    Show that ## 2^{n} ## divides an integer ## N ##.

    Proof: Suppose ## 2^{n} ## divides an integer ## N ##. Let ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##. Then ## 2^{n}\mid N\implies 2^{n}\mid (a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0}) ##. Note that ## 10^{k}=2^{k} 5^{k}\equiv 0\pmod {2^{n}} ##...
  13. M

    No integer whose digits add up to ## 15 ## can be a square or a cube

    Proof: Let ## a ## be any integer. Then ## a\equiv 0, 1, 2, 3, 4, 5, 6, 7 ##, or ## 8\pmod {9} ##. This means ## a^{2}\equiv 0, 1, 4, 9, 7, 7, 0, 4 ##, or ## 1\pmod {9} ## and ## a^{3}\equiv 0, 1, 8, 0, 1, 8, 0, 1 ##, or ## 8\pmod {9} ##. Thus ## a^{2}\equiv 0, 1, 4 ##, or ## 7\pmod {9} ## and...
  14. M

    Proof: Integer Divisibility by 3 via Polynomials

    Proof: Let ## P(x)= \Sigma^{m}_{k=0} a_{k} x^{k} ## be a polynomial function. Then ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##. Since ## 10\equiv 1\pmod {3} ##, it follows that ## P(10)\equiv P(1)\pmod {3} ##. Note that ## N\equiv (a_{m}+a_{m-1}+\dotsb...
  15. M

    Proof That an Integer is Divisible by 2

    Proof: Suppose ## N ## is the integer and ## x ## is the units digit of ## N ##. Then ## N=10k+x ## for some ## k\in\mathbb{Z} ## where ## x={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} ##. Note that ## 10k\equiv 0\pmod {2}\implies N\equiv x\pmod {2} ##. Thus ## 2\mid N\implies N\equiv 0\pmod {2}\implies...
  16. M

    Is the integer ## (447836)_{9} ## divisible by ## 3 ## and ## 8 ##?

    Observe that ## (447836)_{9}=6+3\cdot 9+8\cdot 9^{2}+7\cdot 9^{3}+4\cdot 9^{4}+4\cdot 9^{5}=268224 ##. Then ## 2+6+8+2+2+4=24 ##. Thus ## 3\mid (2+6+8+2+2+4)\implies 3\mid (447836)_{9} ## and ## 8\mid (2+6+8+2+2+4)\implies 8\mid (447836)_{9} ##. Therefore, the integer ## (447836)_{9} ## is...
  17. M

    For any integer ## a ##, the units digit of ## a^{2} ## is?

    Proof: Let ## a ## be any integer. Then ## a\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8 ##, or ## 9\pmod {10} ##. Note that ## a^{2}\equiv 0, 1, 4, 9, 6, 5, 6, 9, 4 ##, or ## 1\pmod {10} ##. Thus ## a^{2}\equiv 0, 1, 4, 5, 6 ##, or ## 9\pmod {10} ##. Therefore, the units digit of ## a^{2} ## is ## 0, 1...
  18. M

    Prove that ## b\equiv c\pmod {n} ##, where the integer is....

    Proof: Suppose ## a\equiv b\pmod {n_{1}} ## and ## a\equiv c\pmod {n_{2}} ## where the integer ## n=gcd(n_{1}, n_{2}) ##. Then ## a-b=n_{1}k_{1} ## and ## a-c=n_{2}k_{2} ## for some ## k_{1}, k_{2}\in\mathbb{Z} ##. This means ## b-c=n_{2}k_{2}-n_{1}k_{1} ##. Since ## n=gcd(n_{1}, n_{2}) ##, it...
  19. H

    Proving that an Integer lies between x and y using Set Theory

    ## y-x \gt 1 \implies y \gt 1+x## Consider the set ##S## which is bounded by an integer ##m##, ## S= \{x+n : n\in N and x+n \lt m\}##. Let's say ##Max {S} = x+n_0##, then we have $$ x+n_0 \leq m \leq x+(n_0 +1)$$ We have, $$ x +n_0 \leq m \leq (x+1) +n_0 \lt y+ n_0 $$ Thus, ##x+n_0 \leq m \lt...
  20. M

    Prove: If the integer ## a ## is not divisible by ## 2 ## or ## 3 ##....

    Proof: Suppose that the integer ## a ## is not divisible by ## 2 ## or ## 3 ##. Then ## a\equiv 1, 5, 7, 11, 13, 17, 19 ## or ## 23\pmod {24} ##. Note that ## a\equiv b\pmod {n}\implies a^{2}\equiv b^{2}\pmod {n} ##. Thus ## a^{2}\equiv 1, 25, 49, 121, 169, 289, 361 ## or ## 529\pmod...
  21. M

    If ## a ## is an odd integer, then ## a^{2}\equiv 1\pmod {8} ##?

    Proof: Suppose ## a ## is an odd integer. Then ## a=2k+1 ## for some ## k\in\mathbb{Z} ##. Note that ## a^{2}=(2k+1)^{2}=4k^{2}+4k+1=4k(k+1)+1 ##. Since ## k(k+1) ## is the product of two consecutive integers, it follows that ## k(k+1) ## must be even. This means ## k(k+1)=2m ## for some ##...
  22. M

    Given an integer, find the probability its cube ends in 111

    The last three digits of ##x^3## must be solely dependent on the last 3 digits of ##x##. So let ##x=a+10b+100c## for integers ##a,b,c##. Then ##x^3 = a^3 + 30 a^2 b + 300 a b^2 + 300 a^2 c +O(1000)## where of course ##O(1000)## don't affect the last 3 digits. Evidently ##a^3## is the only...
  23. M

    Prove that the integer ## 53^{103}+103^{53} ## is divisible by....

    Proof: First, we will prove that the integer ## 53^{103}+103^{53} ## is divisible by ## 39 ##. Note that ## 53\equiv 14 \pmod {39}\implies 53^{2}\equiv 14^{2}\pmod {39}\equiv 196\pmod {39}\equiv 1\pmod {39} ##. Now observe that ## 103\equiv 25\pmod {39}\equiv -14\pmod {39}\implies 103^{2}\equiv...
  24. S

    How many positive integer solutions satisfy this equation?

    $$\frac{1}{x}+\frac{1}{y}+\frac{1}{xy}=\frac{1}{2021}$$ $$\frac{y+x+1}{xy}=\frac{1}{2021}$$ $$xy = 2021y + 2021 x + 2021$$ Then I am stuck. How to continue? Thanks
  25. S

    Find how many points on a circle have an integer distance from other points

    Distance between point (-4, 5) and point on circle: $$d=\sqrt{(x+4)^2+(y-5)^2}$$ $$=\sqrt{x^2+8x+16+y^2-10y+25}$$ Then substitute ##y^2## from equation of circle: $$d=\sqrt{x^2+8x+16-x^2+4x-6y+12-10y+25}$$ $$=\sqrt{12x-16y+53}$$ After this, I need to try the points one by one to check whether...
  26. M

    Are Two Random Numbers More Likely to Have a Quotient Closer to an Odd Integer?

    Closer to odd number implies ##|y/x - (2n+1)| < 1/2## for ##n = 0,1,2...##. Then $$-\frac 1 2 < \frac y x - (2n+1) < \frac 1 2 \implies\\ y < (2n + 1.5)x,\\ y > (2n + 0.5)x$$ for each ##n##. We note ##x \in (0,1)## implies ##y## can be larger than 1 since the slope is greater than 1 (but we know...
  27. M

    Every integer greater than 5 is the sum of three primes?

    Proof: Let ## a>5 ## be an integer. Now we consider two cases. Case #1: Suppose ## a ## is even. Then ## a=2n ## for ## n\geq 3 ##. Note that ## a-2=2n-2=2(n-1) ##, so ## a-2 ## is even. Applying Goldbach's conjecture produces: ## 2n-2=p_{1}+p_{2} ## as a sum of two primes ## p_{1} ## and ##...
  28. M

    Prove that the Goldbach conjecture that every even integer....

    Proof: Let ## n ## be an integer. Then ## 2n=p_{1}+p_{2} ## for ## n\geq 2 ## where ## p_{1} ## and ## p_{2} ## are primes. Suppose ## n=k-1 ## for ## k\geq 3 ##. Then ## 2(k-1)=p_{1}+p_{2} ## ## 2k-2=p_{1}+p_{2} ## ## 2k=p_{1}+p_{2}+2 ##. Thus ## 2k+1=p_{1}+p_{2}+3 ##...
  29. M

    For ## n>1 ##, show that every prime divisor of ## n+1 ## is an odd integer

    Proof: Suppose for the sake of contradiction that there exists a prime divisor of ## n!+1 ##, which is an odd integer that is not greater than ## n ##. Let ## n>1 ## be an integer. Since ## n! ## is even, it follows that ## n!+1 ## is odd. Thus ## 2\nmid (n!+1) ##. This means every prime factor...
  30. M

    If ## a ## is a positive integer and ## \sqrt[n]{a} ## is rational?

    Proof: Suppose ## a ## is a positive integer and ## \sqrt[n]{a} ## is rational. Then we have ## \sqrt[n]{a}=\frac{b}{c} ## for some ## b,c\in\mathbb{Z} ## such that ## gcd(b, c)=1 ## where ## c\neq 0 ##. Thus ## \sqrt[n]{a}=\frac{b}{c} ## ## (\sqrt[n]{a})^{n}=(\frac{b}{c})^n ##...
  31. Mr_Allod

    Edge States in Integer Quantum Hall Effect (IQHE)

    Hello there, I am having trouble understanding what parts b-d of the question are asking. By solving the Schrodinger equation I got the following for the Landau Level energies: $$E_{n,k} = \hbar \omega_H(n+\frac 12)+\frac {\hbar^2k^2}{2m}\frac{\omega^2}{\omega_H^2}$$ Where ##\omega_H =...
  32. M

    Every integer n>1 is the product of a square-free integer?

    Proof: Suppose ## n>1 ## is a positive integer. Let ## n=p_{1}^{k_{1}} p_{2}^{k_{2}}\dotsb p_{r}^{k_{r}} ## be the prime factorization of ## n ## such that each ## k_{i} ## is a positive integer and ## p_{i}'s ## are prime for ## i=1,2,3,...,r ## with ## p_{1}<p_{2}<p_{3}<\dotsb <p_{r} ##...
  33. M

    Determine whether the integer 701 is prime by testing?

    Proof: Consider all primes ## p\leq\sqrt{701}\leq 27 ##. Note that ## 701=2(350)+1 ## ## =3(233)+2 ## ## =5(140)+1 ## ## =7(100)+1 ## ## =11(63)+8 ## ## =13(53)+12 ##...
  34. M

    An integer n>1 is square-free if and only if?

    Proof: Suppose an integer ## n>1 ## is square-free. Then we have ## a^2\nmid n, \forall a\in\mathbb{Z} ##. Let ## n=p_{1}^{a_{1}} p_{2}^{a_{2}}\dotsb p_{r}^{a_{r}} ## be the prime factorization of ## n ## such that each ## a_{i} ## is a positive integer and ## p_{i}'s ## are prime for ##...
  35. M

    Criterion for a positive integer a>1 to be a square

    Proof: Suppose a positive integer ## a>1 ## is a square. Then we have ## a=b^2 ## for some ## b\in\mathbb{Z} ##, where ## b=p_{1}^{n_{1}} p_{2}^{n_{2}} \dotsb p_{r}^{n_{r}} ## such that each ## n_{i} ## is a positive integer and ## p_{i}'s ## are prime for ## i=1,2,3,...,r ## with ##...
  36. M

    Prove n^2+2^n Composite if n not 6k+3

    Proof: Suppose ## n>1 ## is an integer not of the form ## 6k+3 ##. Then we have ## n=6k ## for some ## k\in\mathbb{Z} ##. Thus ## n^{2}+2^{n}=(6k)^{2}+2^{6k} ## ## =36k^{2}+2^{6k} ## ## =2(18k^{2}+2^{6k-1}) ##...
  37. M

    Show that k is an odd integer, except when k=2

    Proof: Suppose for the sake of contradiction that ## p=2^{k}-1 ## is prime but ## k ## is not an odd integer. That is, ## k ## is an even integer. Then we have ## k=2a ## for some ## a\in\mathbb{Z} ##. Thus ## p=2^{k}-1 =2^{2a}-1 =4^{a}-1. ## Note that ## 3\mid...
  38. M

    Each integer n>11 can be written as the sum of two composite numbers?

    Proof: Suppose n is an integer such that ## n>11 ##. Then n is either even or odd. Now we consider these two cases separately. Case #1: Let n be an even integer. Then we have ## n=2k ## for some ## k\in\mathbb{Z} ##. Consider the integer ## n-6 ##. Note...
  39. M

    Any integer of the form ## 8^n+1 ##, where n##\geq##1, is composite?

    Proof: Suppose ##a=8^n+1 ## for some ##a \in\mathbb{Z}## such that n##\geq##1. Then we have ##a=8^n+1 ## =## (2^3)^n+1 ## =## (2^n+1)(2^{2n} -2^n+1) ##. This means ## 2^n+1\mid 2^{3n} +1 ##. Since ##2^n+1>1## and ##2^{2n} -2^n+1>1## for all...
  40. M

    Every integer of the form n^4+4, with n>1, is composite?

    Proof: Suppose a=n^4+4 for some a##\in\mathbb{Z}## such that n>1. Then we have a=n^4+4=(n^2-2n+2)(n^2+2n+2). Note that n^2-2n+2>1 and n^2+2n+2>1 for n>1. Therefore, every integer of the form n^4+4, with n>1, is composite.
  41. M

    Can anyone please review/verify this proof of a nonzero integer a?

    Proof: First, we will show that gcd(a, 0)=abs(a). Suppose a is a nonzero integer such that a##\neq##0. Note that gcd(a, 0)##\le##abs(a) by definition of the greatest common divisor. Since abs(a) divides both a and 0, we have that...
  42. e2m2a

    B Square Root of an Odd Powered Integer is Always Irrational?

    Is it always true that the square root of an odd powered integer will always be irrational?
  43. S

    MHB Find the smallest positive integer N

    Find the smallest positive integer N that satisfies all of the following conditions: • N is a square. • N is a cube. • N is an odd number. • N is divisible by twelve prime numbers. How many digits does this number N have?Please Explain your steps in detail.
  44. Arman777

    Python How does Python tell that an integer is actually an integer?

    Look at this example, >>> a = 97 >>> type(a) <class 'int'> >>> bin(a) '0b1100001' >>> b = ord('a') >>> b 97 >>> type(b) <class 'int'> >>> bin(b) '0b1100001' Is this means that the string 'a' and the integer 97 stored as the same binary in the memory ? If so then how can python tell the...
  45. R

    Compute the number of positive integer divisors of 10

    Compute the number of positive integer divisors of 10!. By the fundamental theorem of arithmetic and the factorial expansion: 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 2 x 5 x 3^2 x 7 x 2 x 3 x 5 x 2^2 x 3 x 2 x 1 = 2^8 x 3^4 x 5^2 x 7 Then there are 9 possibilities for 2, 5 for 3, 3 for...
  46. chwala

    Proving an integer problem about the sum of a square number and a prime number

    i do not seem to understand part ##ii## of this problem...mathematical induction proofs is one area in maths that has always boggled me :oldlaugh: let ##n=3, p=7, ⇒m=4## therefore, ##7=(4-3)(4+3)## ##7=1⋅7## ##1, 7## are integers...##p## is prime. i am attempting part ##iii## in a moment...
  47. A

    I Proving the Finite Binomial Series for k Non-Negative Integer

    Hello, I was wondering how to prove that the Binomial Series is not infinite when k is a non-negative integer. I really don't understand how we can prove this. Do you have any examples that can show that there is a finite number when k is a non-negative integer? Thank you!
  48. anemone

    MHB Positive Integer Solutions of $(x^2+y^2)^n=(xy)^{2014}$

    Find all positive integers $n$ for which the equation $(x^2+y^2)^n=(xy)^{2014}$ has positive integer solutions.
  49. S

    MHB Proving Floor Value of A is an Integer B

    Given the definition of floor value : For all A,B we define the floor value of A denoted by [A] to be an iteger B such that : $[A]=B\Leftrightarrow B\leq A<B+1$ And in symbols $\forall A\forall B ( [A]=B\Leftrightarrow B\leq A<B+1\wedge B\in Z)$,then prove: For all A $ [A]\leq A<[A]+1$
  50. anemone

    MHB Solve Integer & Inequality: $x=(x-1)^3$ for $N$

    Let $x$ be a real number such that $x=(x-1)^3$. Show that there exists an integer $N$ such that $-2^{1000}<x^{2021}-N<2^{-1000}$.
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