Divisibility Rule of 29

Divisibility rules are simple shortcuts that help determine if one number is divisible by another without performing full division. The divisibility rule for 29 helps us quickly determine if a number is divisible by 29 without needing to perform long division.

Divisibility Rule of 29:
Take the number's last digit (unit digit) and multiply it by 3.
Add this value to the rest of the digits in the number.
Check the result: If the number is divisible by 29 (or is 0), then the original number is divisible by 29.

Proof for Divisibility Rule of 29

A general number N can be written as:

N=10^na_n+10^{n−1}a_{n−1}+⋯+10a_1+a_0

Here, a_n,a_n−1,…,a_1,a_0 are the digits of the number. We want to show that N is divisible by 29, i.e., N = 29k for some integer k.

We can factor out 10 from all terms except the last one, giving:

N=10(10^{n−1}a_n+10^{n−2}a_{n−1}+⋯+10a_2+a_1)+a_0

Now, to introduce the rule of subtracting twice the last digit, we add and subtract 30a_0.

N=10(10^{n−1}a_n+10^{n−2}a_{n−1}+⋯+10a_2+a_1)+30a_0−30a_0+a0

This simplifies to:

N=10(10^{n−1}a_n+10^{n−2}a_{n−1}+⋯+10a_2+a_1+3a_0)-29a_0

N+29a_0=10(10^{n−1}a_n+10^{n−2}a_{n−1}+⋯+10a_2+a_1+3a_0)

Now, notice that:

29 ≡ 0 (mod 29) so the term 29a₀ contributes nothing to the remainder of modulo 29.
We only need to check whether 10 (10^{n-1} a_n + 10^{n-2} a_{n-1} + ... + a_1 + 3a_0) ≡ 0 (mod 29)

10 (\overline{a_n + a_{n-1} + ... + a_1} + 3a_0) ≡ 0 (mod 29)

Thus, for N to be divisible by 29, the following must hold:

\overline{a_n + a_{n-1} + ... + a_1} + 3a_0) ≡ 0 (mod 29)

More Examples of Divisibility by 29 Rule

Here are a few examples of numbers divisible by 29, applying the divisibility rule:

Example 1: Check if 1530 is divisible by 29

Solution:

Take the last digit (0) and multiply it by 3 to get 0.
Add it to the remaining number (153): 153 + 0 = 153.
Since 153 is still large, apply the rule again:
Take the last digit (3) of 153 and multiply it by 3 to get 9.
Add it to the remaining number (15): 15 + 9 = 24.
Check if 24 is divisible by 29: 24 ÷ 29 ≠ whole number.
Since 24 is not divisible by 29, 1530 is not divisible by 29.

Example 2: Check if 3567 is divisible by 29

Solution:

Take the last digit (7) and multiply it by 3 to get 21.
Add it to the remaining number (356): 356 + 21 = 377.
Since 377 is still large, apply the rule again:
Take the last digit (7) of 377 and multiply it by 3 to get 21.
Add it to the remaining number (37): 37 + 21 = 58.
Check if 58 is divisible by 29: 58 ÷ 29 2 (whole number).
Since 58 is divisible by 29, 3567 is also divisible by 29.