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Chapter 1: Problem 2

Find all divisors of the given number. $$19$$

Short Answer

Expert verified
The divisors of 19 are 1 and 19.

Step by step solution

01

Understand Divisors

Divisors of a number are integers that can divide the number without leaving a remainder. We want to find these for the number 19.
02

Identify if 19 is Prime

A prime number is one with exactly two divisors: 1 and itself. To check if 19 is prime, we see whether there are no divisors other than 1 and 19 between 1 and itself.
03

Check for Divisors

Start checking numbers from 2 to the square root of 19 (approximately 4.36) to see if any divide 19 without a remainder. Numbers 2, 3, and 4 do not divide 19 cleanly.
04

Determine the Divisors

Since no numbers other than 1 and 19 divide 19 without a remainder, the divisors of 19 are 1 and 19 itself.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

prime numbers
A prime number is a special kind of number in mathematics. It's a number that can only be divided exactly by itself and 1. This means when you divide a prime number by any other numbers, you will get a remainder. For instance, let's consider the number 19.
  • 19 is only divisible by 1 and 19 without leaving a remainder.
  • It cannot be divided evenly by 2, 3, or 4 because it would leave a remainder.
  • Therefore, 19 is a prime number.
The idea is to check for divisibility using numbers less than or equal to the square root of 19. But since none of those smaller numbers can divide it evenly, it confirms that 19 is indeed prime.
integer division
Integer division is when you divide one whole number by another and result in a quotient that is also a whole number, with any remainder being discarded. It does not work like regular division where we might end up with a fraction or decimal.
Consider dividing 19 by 3:
  • The result of dividing 19 by 3 is 6, with a remainder of 1.
  • In integer division, you'd say that 19 divided by 3 equals 6.
  • The remainder is not expressed in the quotient.
When seeking divisors, you perform integer division to find out if a number divides evenly into another. If the division result yields a whole number without a remainder, that number is a divisor.
remainder
The remainder is what鈥檚 left over after performing an integer division. It helps us understand whether a number is a divisor of another.
  • If you divide 19 by any number and get a remainder of 0, that number is a divisor.
  • For 19, dividing by numbers like 2, 3, or 4 leaves a remainder, proving they are not divisors.
  • Only 1 and 19 divide it cleanly, leaving no remainder.
The concept of remainder is crucial when checking for numbers that perfectly divide another number. It essentially filters out non-divisors by the presence of a leftover amount.

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