Signs of the divisibility of numbers

In this publication, we will consider the signs of divisibility by numbers from 2 to 11, accompanying them with examples for a better understanding.

Certificate of divisibility – this is an algorithm, using which you can relatively quickly determine whether the number under consideration is a multiple of a predetermined one (that is, whether it is divisible by it without a remainder).

Sign of divisibility on 2

A number is divisible by 2 if and only if its last digit is even, i.e. is also divisible by two.

examples:

• 4, 32, 50, 112, 2174 – the last digits of these numbers are even, which means they are divisible by 2.
• 5, 11, 37, 53, 123, 1071 – are not divisible by 2, because their last digits are odd.

Sign of divisibility on 3

A number is divisible by 3 if and only if the sum of all its digits is also divisible by XNUMX.

examples:

• 18 – divisible by 3, because. 1+8=9, and the number 9 is divisible by 3 (9:3=3).
• 132 – divisible by 3, because. 1+3+2=6 and 6:3=2.
• 614 is not a multiple of 3, because 6+1+4=11, and 11 is not evenly divisible by 3 (11: 3 = 3²/₃).

Sign of divisibility on 4

two digit number

A number is divisible by 4 if and only if the sum of twice the digit in its tens place and the digit in the ones place is also divisible by four.

examples:

• 64 – divisible by 4, because. 6⋅2+4=16 and 16:4=4.
• 35 is not divisible by 4, because 3⋅2+5=11, and 11: 4 2 =³/₄.

Number of digits greater than 2

A number is a multiple of 4 when its last two digits form a number divisible by four.

examples:

• 344 – divisible by 4, because. 44 is a multiple of 4 (according to the algorithm above: 4⋅2+4=12, 12:4=3).
• 5219 is not a multiple of 4, because 19 is not divisible by 4.

Note:

A number is divisible by 4 without a remainder if:

• in its last digit are the numbers 0, 4 or 8, and the penultimate digit is even;
• in the last digit – 2 or 6, and in the penultimate – odd numbers.

Sign of divisibility on 5

A number is divisible by 5 if and only if its last digit is 0 or 5.

examples:

• 10, 65, 125, 300, 3480 – divisible by 5, because end in 0 or 5.
• 13, 67, 108, 649, 16793 – are not divisible by 5, because their last digits are not 0 or 5.

Sign of divisibility on 6

A number is divisible by 6 if and only if it is a multiple of both two and three at the same time (see signs above).

examples:

• 486 – divisible by 6, because. is divisible by 2 (the last digit of 6 is even) and by 3 (4+8+6=18, 18:3=6).
• 712 – not divisible by 6, because it is only a multiple of 2.
• 1345 – not divisible by 6, because is not a multiple of either 2 or 3.

Sign of divisibility on 7

A number is divisible by 7 if and only if the sum of three times its tens and the digits in the ones place is also divisible by seven.

examples:

• 91 – divisible by 7, because. 9⋅3+1=28 and 28:7=4.
• 105 – divisible by 7, because. 10⋅3+5=35, and 35:7=5 (in the number 105 there are ten tens).
• 812 is divisible by 7. Here the following chain is: 81⋅3+2=245, 24⋅3+5=77, 7⋅3+7=28, and 28:7=4.
• 302 – not divisible by 7, because 30⋅3+2=92, 9⋅3+2=29, and 29 is not divisible by 7.

Sign of divisibility on 8

three digit number

A number is divisible by 8 if and only if the sum of the digit in the ones place, twice the digit in the tens place, and quadruple the digit in the hundreds place is divisible by eight.

examples:

• 264 – divisible by 8, because. 2⋅4+6⋅2+4=24 and 24:8=3.
• 716 – 8 is not divisible, because 7⋅4+1⋅2+6=36, and 36: 8 4 =¹/₂.

Number of digits greater than 3

A number is divisible by 8 when the last three digits form a number divisible by 8.

examples:

• 2336 – divisible by 8, because 336 is a multiple of 8.
• 12547 is not a multiple of 8, because 547 is not evenly divisible by eight.

Sign of divisibility on 9

A number is divisible by 9 if and only if the sum of all its digits is also divisible by nine.

examples:

• 324 – divisible by 9, because. 3+2+4=9 and 9:9=1.
• 921 – not divisible by 9, because 9+2+1=12 and 12: 9 1 =¹/₃.

Sign of divisibility on 10

A number is divisible by 10 if and only if it ends in zero.

examples:

• 10, 110, 1500, 12760 are multiples of 10, the last digit is 0.
• 53, 117, 1254, 2763 are not divisible by 10.

Sign of divisibility on 11

A number is divisible by 11 if and only if the difference between the sums of even and odd digits is zero or divisible by eleven.

examples:

• 737 – divisible by 11, because. |(7+7)-3|=11, 11:11=1.
• 1364 – divisible by 11, because |(1+6)-(3+4)|=0.
• 24587 is not divisible by 11 because |(2+5+7)-(4+8)|=2 and 2 is not divisible by 11.