Fix the mistake: Properties of Numbers

Properties of Numbers ① Basics of Multiples and Divisors

In this lesson, you will learn the meanings of “multiples” and “divisors.” This is an important topic that frequently appears on junior high school entrance exams, and it is the foundation for the important concepts of “greatest common divisor” and “least common multiple,” which will be covered later.

What You Will Learn

A multiple is a number obtained by multiplying an integer (a number that is not a decimal or fraction) by 1, 2, 3, and so on. For example, the multiples of 3 continue as 3, 6, 9, 12, 15, and so on.
A divisor is an integer that divides another integer evenly. For example, the divisors of 12 are the numbers that divide 12 evenly: 1, 2, 3, 4, 6, and 12, for a total of 6.

Example Problem
Find all the divisors of 24.

Solution:
① Write down, in order from the smallest numbers, the pairs of numbers whose product is 24.
　1×24, 2×12, 3×8, 4×6
② Arrange all the numbers obtained in ascending order.
Answer: The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24, for a total of 8.

Key Points

Remember
  • When looking for divisors, think in terms of “multiplication pairs” so that you do not miss any.
  • Multiples continue without end, but the number of divisors is finite.
  • For any integer, 1 and the number itself are always divisors.

Practice Problems

  1. Write all the divisors of 36.
  2. Write the first five multiples of 7 in ascending order.
  3. Among the integers from 1 to 30, find all the numbers that have exactly 3 divisors.

Answers and Explanations

  1. Answer:1, 2, 3, 4, 6, 9, 12, 18, 36 (9 numbers)
    Explanation: Think in terms of multiplication pairs. Listing the numbers in 1×36, 2×18, 3×12, 4×9, and 6×6 ensures that none are missed.
  2. Answer:7, 14, 21, 28, 35
    Explanation: The numbers obtained by multiplying 7 by 1, 2, 3, 4, and 5 are multiples of 7.
  3. Answer:4, 9, 25 (numbers in the form of the square of a prime number)
    Explanation: A number has exactly 3 divisors when it is in the form “p² (where p is prime).” From 1 to 30, these are 2²=4, 3²=9, and 5²=25. (The three divisors are 1, p, and p².)

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