kb/data/en.wikipedia.org/wiki/Divisibility_rule-3.md

title	chunk	source	category	tags	date_saved	instance
Divisibility rule	4/8	https://en.wikipedia.org/wiki/Divisibility_rule	reference	science, encyclopedia	2026-05-05T08:13:56.367542+00:00	kb-cron

For all of the above examples, subtracting the first three digits from the last three results in a multiple of seven. Notice that leading zeros are permitted to form a 6-digit pattern. This phenomenon forms the basis for Steps B and C. Step B: If the integer is between 1001 and one million, find a repeating pattern of 1, 2, or 3 digits that forms a 6-digit number that is close to the integer (leading zeros are allowed and can help you visualize the pattern). If the positive difference is less than 1000, apply Step A. This can be done by subtracting the first three digits from the last three digits. For example:

341,355 − 341,341 = 14 -> 1 − (4×2) = 1 − 8 = −7 YES 67,326 − 067,067 = 259 -> 25 − (9×2) = 25 − 18 = 7 YES

The fact that 999,999 is a multiple of 7 can be used for determining divisibility of integers larger than one million by reducing the integer to a 6-digit number that can be determined using Step B. This can be done easily by adding the digits left of the first six to the last six and follow with Step A. Step C: If the integer is larger than one million, subtract the nearest multiple of 999,999 and then apply Step B. For even larger numbers, use larger sets such as 12-digits (999,999,999,999) and so on. Then, break the integer into a smaller number that can be solved using Step B. For example:

22,862,420 − (999,999 × 22) = 22,862,420 − 21,999,978 -> 862,420 + 22 = 862,442 862,442 -> 862 − 442 (Step B) = 420 -> 42 − (0×2) (Step A) = 42 YES

This allows adding and subtracting alternating sets of three digits to determine divisibility by seven. Understanding these patterns allows you to quickly calculate divisibility of seven as seen in the following examples: Pohlman–Mass method of divisibility by 7, examples:

Is 98 divisible by seven? 98 -> 9 − (8×2) = 9 − 16 = −7 YES (Step A)

Is 634 divisible by seven? 634 -> 63 − (4×2) = 63 − 8 = 55 NO (Step A)

Is 355,341 divisible by seven? 355,341 − 341,341 = 14,000 (Step B) -> 014 − 000 (Step B) -> 14 = 1 − (4×2) (Step A) = 1 − 8 = −7 YES

Is 42,341,530 divisible by seven? 42,341,530 -> 341,530 + 42 = 341,572 (Step C) 341,572 − 341,341 = 231 (Step B) 231 -> 23 − (1×2) = 23 − 2 = 21 YES (Step A)

Using quick alternating additions and subtractions: 42,341,530 -> 530 − 341 + 42 = 189 + 42 = 231 -> 23 − (1×2) = 21 YES

Multiplication by 3 method of divisibility by 7, examples:

Is 98 divisible by seven? 98 -> 9 remainder 2 -> 2×3 + 8 = 14 YES

Is 634 divisible by seven? 634 -> 6×3 + 3 = 21 -> remainder 0 -> 0×3 + 4 = 4 NO

Is 355,341 divisible by seven? 3 × 3 + 5 = 14 -> remainder 0 -> 0×3 + 5 = 5 -> 5×3 + 3 = 18 -> remainder 4 -> 4×3 + 4 = 16 -> remainder 2 -> 2×3 + 1 = 7 YES

Find remainder of 1036125837 divided by 7 1×3 + 0 = 3 3×3 + 3 = 12 remainder 5 5×3 + 6 = 21 remainder 0 0×3 + 1 = 1 1×3 + 2 = 5 5×3 + 5 = 20 remainder 6 6×3 + 8 = 26 remainder 5 5×3 + 3 = 18 remainder 4 4×3 + 7 = 19 remainder 5 Answer is 5