Divisors of a number are the integers that can divide that number without leaving any remainder. In simpler terms, a divisor is a number that you can use to divide another number evenly. For example, if a number can be split into equal parts without having anything left over, then that number is a divisor. The purpose of this article is to help you understand how to find the divisors of a specific number, using the number 407 as our example.
Understanding Division
Before we find the divisors of 407, let’s break down the concept of division. When you divide one number by another, you have three main terms to know:
- Dividend: This is the number you want to divide. In our case, the dividend is 407.
- Divisor: This is the number you are dividing by. We will test each integer to see if it can divide 407.
- Quotient: This is the result of the division. It tells you how many times the divisor fits into the dividend.
For example, let’s look at how dividing 407 by 1 works:
407 / 1 = 407
In this case:
- The dividend is 407
- The divisor is 1
- The quotient is 407
Step-by-Step Guide to Calculating Divisors
Now, let’s figure out the divisors of 407. We’ll start by dividing 407 by every integer from 1 up to 407 itself. If the answer (quotient) is a whole number, then that integer is a divisor of 407.
- First, start with 1 and go up to 407:
- Divide 407 by each integer and check for whole numbers:
Let’s go through some calculations:
- 407 / 1 = 407 (whole number, so 1 is a divisor)
- 407 / 2 = 203.5 (not a whole number)
- 407 / 3 = 135.666… (not a whole number)
- 407 / 4 = 101.75 (not a whole number)
- 407 / 5 = 81.4 (not a whole number)
- 407 / 6 = 67.833… (not a whole number)
- 407 / 7 = 58.142857… (not a whole number)
- 407 / 8 = 50.875 (not a whole number)
- 407 / 9 = 45.2222… (not a whole number)
- 407 / 10 = 40.7 (not a whole number)
Now, let’s go further to see if we can find more divisors at higher numbers:
- 407 / 11 = 37.9090… (not a whole number)
- 407 / 12 = 33.91666… (not a whole number)
- 407 / 13 = 31.3076… (not a whole number)
- 407 / 14 = 29.0714 (not a whole number)
- 407 / 15 = 27.1333… (not a whole number)
- 407 / 16 = 25.4375 (not a whole number)
- 407 / 17 = 23.9411… (not a whole number)
- 407 / 18 = 22.6111… (not a whole number)
- 407 / 19 = 21.3684 (not a whole number)
- 407 / 20 = 20.35 (not a whole number)
- 407 / 21 = 19.3809… (not a whole number)
- 407 / 22 = 18.5 (not a whole number)
- 407 / 23 = 17.6521739… (not a whole number)
- 407 / 24 = 16.9583… (not a whole number)
- 407 / 25 = 16.28 (not a whole number)
- 407 / 26 = 15.5769…(not a whole number)
- 407 / 27 = 15.0370…(not a whole number)
- 407 / 28 = 14.5357…(not a whole number)
- 407 / 29 = 14.0344…(not a whole number)
- 407 / 30 = 13.5667…(not a whole number)
- 407 / 31 = 13.129… (not a whole number)
- 407 / 32 = 12.71875…(not a whole number)
- 407 / 33 = 12.3030…(not a whole number)
- 407 / 34 = 11.9765…(not a whole number)
- 407 / 35 = 11.628571…(not a whole number)
- 407 / 36 = 11.305555…(not a whole number)
- 407 / 37 = 11 (whole number, so 37 is a divisor)
- 407 / 38 = 10.6842…(not a whole number)
- 407 / 39 = 10.4359…(not a whole number)
- 407 / 40 = 10.175…(not a whole number)
- 407 / 41 = 9.902439…(not a whole number)
Continuing this process reveals:
- The whole numbers we found that divided evenly into 407 were 1 and 37.
Conclusion
The complete list of divisors for the number 407 is:
- 1
- 37
Understanding divisors helps us in many areas of math, especially in fraction simplification and determining factors in number theory. Finding the divisors of a number can also be useful in various real-life applications, such as sharing items evenly among a group.