"The attempt to make piano arithmetic be about the integers is that the set of relations defined by piano arithmetic sufficiently kind of defining that you get the integers and nothing but the integers."
"There are no positive integers such that p and Q such that 4p squared minus Q squared equals 25."
"We can use built-in facilities like data type to check how, when we run that, we're working with integers."
"When you multiply two negative integers, you'll get a positive answer."
"In order to be a rational number, there has to be an integer."
"Most integers have an odd number of prime factors."
"At their heart, all rings are generalizations of the integers."
"The equation X to the N plus Y to the N equals Z to the N has no solution in integers for n greater than or equal to 3 if X, Y, and Z are non-zero."
"God made the integers, all the rest is the work of man."
"Integers can contain positive, negative whole numbers, and 0."
"The integers are set theoretically isomorphic to the natural numbers; there are as many integers as the natural numbers."
"Perfect squares are numbers like 4, 9, 16, 25, etc., because when I take the square root of these numbers, you get a lovely integer value."
"7 is prime in the set of integers because it cannot be factored into a product of smaller numbers in Z."
"A perfect square is a number that is an integer times itself."
"These are the only numbers that you can take a square root of and it gives you a whole number answer."
"For all non-zero integers a and b, there are integers x and y such that ax + by equals gcd(a, b)."
"Modular arithmetic is a system of arithmetic for integers where numbers wrap around a certain value, which is called the modulus."
"Any integer is either odd or even, it is impossible for an integer to be both odd and even or neither odd nor even."
"The floor of X is the greatest integer less than or equal to X."
"The ceiling of X is the least integer greater than or equal to X."
"The sum of two integers is an integer."
"Integers are basically numbers without a decimal point; floats are numbers with a decimal point."
"We have found a solution and that is the solution K equals 3, L equals 3, M equals 4."
"Integers basically are going to be negative and positive whole numbers."
"When you're counting the number of bracelets made, you would count in whole numbers."
"We're only looking at whole numbers here."
"If I add two integers, I always know I get another integer, so I know this operation is closed under addition."
"Integers are whole numbers either positive or negative that have no fractional part."
"The difference between the squares of any two consecutive integers is always an odd number."
"Anytime we're dealing with consecutive integers, please understand that we're just talking about numbers that follow one another."
"The three consecutive integers that add to 84 are 27, 28, and 29."
"My three consecutive integers whose sum of the least and greatest would be equal to 32 more than the middle integer would be 31, 32, and 33."
"We need to find all the numbers a and b out of the positive integers including zero satisfying this: square root a plus square root of b is equal to square root of 2009."
"For every positive integer N, we define ZN to be the integers zero through N minus one."
"Now among the elements of ZN for any positive N, some of them will have GCD equal to one with N and some won't."
"If you've got two integers, x and y, x is either less than y, greater than y, or equal to y."
"Integers [are] a positive, negative, or zero number with no decimal."
"It takes all the multiples of a given number and sews together the integers from a line into a big loop, a big circle."
"Modular arithmetic deals with wrapping an integer value around a fixed value."
"If you're using int + int, should not have decimal points; those would be real numbers instead."
"Integers are nice in that they have what we call infinite precision."