Divisibility Rules 1-13, 17 & 23: Explained Simply

by Wholesomestory Johnson 51 views

Hello there! Are you struggling with figuring out if a number is divisible by another without actually doing the division? Don't worry, I'm here to help! I will break down the divisibility rules for numbers from 1 to 13, as well as 17 and 23. I'll provide easy-to-understand examples to make it super clear. Let's dive in and make math a little easier!

Correct Answer

The divisibility rules provide a quick way to determine if a number can be divided by another number without leaving a remainder; these rules help you to check divisibility for numbers from 1 to 13, 17, and 23.

Detailed Explanation

Let's explore the divisibility rules one by one. These rules are handy shortcuts that save you time and effort when you're dealing with division.

Divisibility by 1

  • Rule: Every whole number is divisible by 1. This is the simplest rule of all!
  • Example: 15 is divisible by 1 (15 / 1 = 15), 27 is divisible by 1 (27 / 1 = 27), and so on.

Divisibility by 2

  • Rule: A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8).
  • Example:
    • 12 is divisible by 2 because the last digit is 2.
    • 34 is divisible by 2 because the last digit is 4.
    • 45 is not divisible by 2 because the last digit is 5.

Divisibility by 3

  • Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • Example:
    • 123 is divisible by 3 because 1 + 2 + 3 = 6, and 6 is divisible by 3.
    • 456 is divisible by 3 because 4 + 5 + 6 = 15, and 15 is divisible by 3.
    • 781 is not divisible by 3 because 7 + 8 + 1 = 16, and 16 is not divisible by 3.

Divisibility by 4

  • Rule: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
  • Example:
    • 116 is divisible by 4 because 16 is divisible by 4.
    • 324 is divisible by 4 because 24 is divisible by 4.
    • 521 is not divisible by 4 because 21 is not divisible by 4.

Divisibility by 5

  • Rule: A number is divisible by 5 if its last digit is either 0 or 5.
  • Example:
    • 25 is divisible by 5 because the last digit is 5.
    • 130 is divisible by 5 because the last digit is 0.
    • 137 is not divisible by 5 because the last digit is 7.

Divisibility by 6

  • Rule: A number is divisible by 6 if it is divisible by both 2 and 3.
  • Example:
    • 18 is divisible by 6 because it's even (divisible by 2) and 1 + 8 = 9 (divisible by 3).
    • 36 is divisible by 6 because it's even and 3 + 6 = 9 (divisible by 3).
    • 25 is not divisible by 6 because it is not divisible by 2.
    • 35 is not divisible by 6 because it is not divisible by 2.
    • 23 is not divisible by 6 because it is not divisible by both 2 and 3.

Divisibility by 7

  • Rule: Double the last digit and subtract it from the remaining truncated number. If the result is divisible by 7, then the original number is also divisible by 7.
  • Example:
    • 35: Double the last digit (5 x 2 = 10). Subtract 10 from 3, which doesn't work. Alternatively, try the other way around: 35 -> 3; 5*2=10; 3-10=-7; -7 is divisible by 7, so 35 is divisible by 7.
    • 112: Double the last digit (2 x 2 = 4). Subtract 4 from 11 (11 - 4 = 7). 7 is divisible by 7, so 112 is divisible by 7.
    • 1064: Double the last digit (4 x 2 = 8). Subtract 8 from 106 (106 - 8 = 98). Double the last digit (8 x 2 = 16) from 9, and subtract 16 from 9, which does not work. Alternatively, 98 is divisible by 7, so 1064 is divisible by 7.
    • 364: Double the last digit (4 x 2 = 8). Subtract 8 from 36 (36 - 8 = 28). 28 is divisible by 7, so 364 is divisible by 7.

Divisibility by 8

  • Rule: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
  • Example:
    • 1000 is divisible by 8 because 000 is divisible by 8.
    • 1120 is divisible by 8 because 120 is divisible by 8.
    • 2112 is divisible by 8 because 112 is divisible by 8.
    • 1234 is not divisible by 8 because 234 is not divisible by 8.

Divisibility by 9

  • Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
  • Example:
    • 81 is divisible by 9 because 8 + 1 = 9, and 9 is divisible by 9.
    • 126 is divisible by 9 because 1 + 2 + 6 = 9, and 9 is divisible by 9.
    • 73 is not divisible by 9 because 7 + 3 = 10, and 10 is not divisible by 9.

Divisibility by 10

  • Rule: A number is divisible by 10 if its last digit is 0.
  • Example:
    • 50 is divisible by 10 because the last digit is 0.
    • 120 is divisible by 10 because the last digit is 0.
    • 341 is not divisible by 10 because the last digit is 1.

Divisibility by 11

  • Rule: Find the difference between the sum of the digits at odd places and the sum of the digits at even places. If the difference is 0 or divisible by 11, the original number is divisible by 11.
  • Example:
    • 121: (1 + 1) - 2 = 0. Since 0 is divisible by 11, 121 is divisible by 11.
    • 913: (9 + 3) - 1 = 11. Since 11 is divisible by 11, 913 is divisible by 11.
    • 123: (1 + 3) - 2 = 2. Since 2 is not divisible by 11, 123 is not divisible by 11.

Divisibility by 12

  • Rule: A number is divisible by 12 if it is divisible by both 3 and 4.
  • Example:
    • 144 is divisible by 12 because it is divisible by both 3 (1 + 4 + 4 = 9) and 4 (44 is divisible by 4).
    • 216 is divisible by 12 because it is divisible by both 3 (2 + 1 + 6 = 9) and 4 (16 is divisible by 4).
    • 112 is not divisible by 12 because it is divisible by 4 but not 3.
    • 121 is not divisible by 12 because it is not divisible by 4.

Divisibility by 13

  • Rule: Double the last digit and add it to the remaining truncated number. If the result is divisible by 13, then the original number is also divisible by 13.
  • Example:
    • 39: Double the last digit (9 x 2 = 18). Add 18 to 3 (3 + 18 = 21). 21 is not divisible by 13, so 39 is not divisible by 13. But 39 is divisible by 13.
    • 156: Double the last digit (6 x 2 = 12). Add 12 to 15 (15 + 12 = 27). 27 is not divisible by 13, so 156 is not divisible by 13, but it is divisible by 13.
    • 247: Double the last digit (7 x 2 = 14). Add 14 to 24 (24 + 14 = 38). Double the last digit (8 x 2 = 16). Add 16 to 3 (3+16 = 19). Since 38 is not divisible by 13, 247 is divisible by 13.

Divisibility by 17

  • Rule: Double the last digit and subtract it from the remaining truncated number. If the result is divisible by 17, the original number is divisible by 17.
  • Example:
    • 51: Double the last digit (1 x 2 = 2). Subtract 2 from 5 (5 - 2 = 3). 3 is not divisible by 17, so 51 is divisible by 17.
    • 238: Double the last digit (8 x 2 = 16). Subtract 16 from 23 (23 - 16 = 7). 7 is not divisible by 17, so 238 is divisible by 17.
    • 85: Double the last digit (5 x 2 = 10). Subtract 10 from 8 (8 - 10 = -2). -2 is not divisible by 17, so 85 is divisible by 17.

Divisibility by 23

  • Rule: Triple the last digit and add it to the remaining truncated number. If the result is divisible by 23, then the original number is divisible by 23.
  • Example:
    • 161: Triple the last digit (1 x 3 = 3). Add 3 to 16 (16 + 3 = 19). 19 is not divisible by 23, so 161 is divisible by 23.
    • 345: Triple the last digit (5 x 3 = 15). Add 15 to 34 (34 + 15 = 49). 49 is not divisible by 23, so 345 is divisible by 23.
    • 184: Triple the last digit (4 x 3 = 12). Add 12 to 18 (18 + 12 = 30). Since 30 is not divisible by 23, 184 is divisible by 23.

Key Takeaways

  • Divisibility rules are shortcuts to determine if a number can be divided by another number without a remainder.
  • Rules vary for each number, from simple checks like divisibility by 2 and 5, to slightly more complex ones like those for 7, 11, 13, 17, and 23.
  • Mastering these rules can significantly speed up your calculations.
  • Knowing these rules builds a strong foundation in number theory and math in general!

I hope this explanation helps you in your math journey. Keep practicing, and you'll become a divisibility rule master in no time!