Divisibility Rule of 171

The divisibility rule for 171 is a method by which we can find out if a number is divisible by 171 or not without using the division method. Check whether 3420 is divisible by 171 with the divisibility rule.
 

Step 1: Check if the number is divisible by both 9 and 19, because the prime factorization of 171 is 9 × 19.
 

To check divisibility by 9, sum the digits of the number. If the sum is a multiple of 9, then the number is divisible by 9. For 3420, the sum of the digits is 3+4+2+0=9, which is a multiple of 9.
 

To check divisibility by 19, use the divisibility rule for 19: Multiply the last digit by 2, subtract this from the remaining leading truncated number, and check if the result is a multiple of 19. For 3420, multiply 0 by 2, which is 0. Subtract 0 from 342, so 342. Now, 342 ÷ 19 = 18, which is a whole number, so it is divisible by 19.
 

Step 2: Since 3420 is divisible by both 9 and 19, it is also divisible by 171.

Knowing the divisibility rules for 9 and 19 will help you master the rule for 171.
 

• Know the multiples of 171: Memorize the multiples of 171 (171, 342, 513, 684, etc.) to quickly check divisibility.
   
• Use the sum of digits: Quickly check if the sum of the digits is divisible by 9.
   
• Repeat the process for large numbers: If the number is large, you can break down the checks to ensure accuracy.
  
  For example, check if 6849 is divisible by 171. First, check for divisibility by 9: 6+8+4+9=27, which is divisible by 9.
  
  Then, for divisibility by 19: Multiply the last digit by 2 (9×2=18), subtract from the truncated number (684-18=666), and check if 666 is divisible by 19.
  
  Since 666 ÷ 19 = 35.05, it is not divisible, thus 6849 is not divisible by 171.
   
• Use the division method to verify: Verify results by dividing the number by 171 to confirm your result.
   

• Divisibility Rule: A set of guidelines to determine if a number is divisible by another without performing division.
   
• Multiple: A number that can be divided by another number without leaving a remainder.
   
• Sum of Digits: The total obtained by adding all the digits of a number.
   
• Prime Factorization: Breaking down a number into its basic building blocks, or prime numbers.
   
• Integer: A whole number, positive or negative, including zero.