Divisibility rule: Difference between revisions

→‎Divisibility rules for numbers 1–20: add "id" attributes to table for easier linking
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(→‎Divisibility rules for numbers 1–20: add "id" attributes to table for easier linking)
Note: To test divisibility by any number that can be expressed as the product of prime factors <math>p_1^n p_2^m p_3^q</math>, we can separately test for divisibility by each prime to its appropriate power. For example, testing divisibility by 18 (18 = 9*2 = 3<sup>2</sup>*2) is equivalent to testing divisibility by 9 (3<sup>2</sup>) and 2 simultaneously, thus we need only show divisibility by 9 and by 2 to prove divisibility by 18.
 
<!-- Note: the "id" attributes are there to allow direct linking to this table as e.g. [[Divisibility rule#7]]. -->
{| class="wikitable"
! Divisor
! Examples
|-
|id=1| '''[[1 (number)|1]]'''
| No special condition. Any integer is divisible by 1.
| 2 is divisible by 1.
|-
|id=2| '''[[2 (number)|2]]'''
| The last digit is even (0, 2, 4, 6, or 8).<ref name="Pascal's-criterion">This follows from Pascal's criterion. See Kisačanin (1998), [{{Google books|plainurl=y|id=BFtOuh5xGOwC|page=101|text=A number is divisible by}} p. 100–101]</ref><ref name="last-m-digits">A number is divisible by 2<sup>''m''</sup>, 5<sup>''m''</sup> or 10<sup>''m''</sup> if and only if the number formed by the last ''m'' digits is divisible by that number. See Richmond & Richmond (2009), [{{Google books|plainurl=y|id=HucyKYx0_WwC|page=105|text=formed by the last}} p. 105]</ref>
| 1,294: 4 is even.
|-
|id=3 rowspan=2| '''[[3 (number)|3]]'''
| Sum the digits. If the result is divisible by 3, then the original number is divisible by 3.<ref name="Pascal's-criterion"/><ref name="apostol-1976">Apostol (1976), [{{Google books|plainurl=y|id=Il64dZELHEIC|page=108|text=sum of its digits}} p. 108]</ref><ref name="Richmond-Richmond-2009">Richmond & Richmond (2009), [{{Google books|plainurl=y|id=HucyKYx0_WwC|page=102|text=divisible by}} Section 3.4 (Divisibility Tests), p. 102–108]</ref>
| 405 → 4 + 0 + 5 = 9 and 636 → 6 + 3 + 6 = 15 which both are clearly divisible by 3.<br>16,499,205,854,376 → 1+6+4+9+9+2+0+5+8+5+4+3+7+6 sums to 69 → 6 + 9 = 15 → 1 + 5 = 6, which is clearly divisible by 3.
| Using the example above: 16,499,205,854,376 has '''four''' of the digits 1, 4 and 7 and '''four''' of the digits 2, 5 and 8; ∴ Since 4 − 4 = 0 is a multiple of 3, the number 16,499,205,854,376 is divisible by 3.
|-
|id=4 rowspan=3| '''[[4 (number)|4]]'''
| Examine the last two digits.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
| 40832: 32 is divisible by 4.
| 40832: 2 × 3 + 2 = 8, which is divisible by 4.
|-
|id=5| '''[[5 (number)|5]]'''
| The last digit is 0 or 5.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
| 495: the last digit is 5.
|-
|id=6| '''[[6 (number)|6]]'''
| It is divisible by 2 and by 3.<ref name="product-of-coprimes">Richmond & Richmond (2009), [{{Google books|plainurl=y|id=HucyKYx0_WwC|page=102|text=divisible by the product}} Section 3.4 (Divisibility Tests), Theorem 3.4.3, p. 107]</ref>
| 1,458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6.
|-
|id=7 rowspan=5| '''[[7 (number)|7]]'''
| Form the [[alternating sum]] of blocks of three from right to left.<ref name="Richmond-Richmond-2009"/><ref name="alternating-sum-of-blocks-of-three">Kisačanin (1998), [{{Google books|plainurl=y|id=BFtOuh5xGOwC|page=101|text=third criterion for 11}} p. 101]</ref>
| 1,369,851: 851 − 369 + 1 = 483 = 7 × 69
| 483595: (4 × (-2)) + (8 × (-3)) + (3 × (-1)) + (5 × 2) + (9 × 3) + (5 × 1) = 7.
|-
|id=8 rowspan=5| '''[[8 (number)|8]]'''
|style="border-bottom: hidden;"| If the hundreds digit is even, examine the number formed by the last two digits.
|style="border-bottom: hidden;"| 624: 24.
| 34152: 4 × 1 + 5 × 2 + 2 = 16
|-
|id=9| '''[[9 (number)|9]]'''
| Sum the digits. If the result is divisible by 9, then the original number is divisible by 9.<ref name="Pascal's-criterion"/><ref name="apostol-1976"/><ref name="Richmond-Richmond-2009"/>
| 2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9.
|-
|id=10| '''[[10 (number)|10]]'''
| The last digit is 0.<ref name="last-m-digits"/>
| 130: the last digit is 0.
|-
|id=11 rowspan=6| '''[[11 (number)|11]]'''
| Form the alternating sum of the digits.<ref name="Pascal's-criterion"/><ref name="Richmond-Richmond-2009"/>
| 918,082: 9 − 1 + 8 − 0 + 8 − 2 = 22.
| 14,179: the number of digits is odd (5) → 417 − 1 − 9 = 407 = 37 × 11
|-
|id=12 rowspan=2| '''[[12 (number)|12]]'''
| It is divisible by 3 and by 4.<ref name="product-of-coprimes"/>
| 324: it is divisible by 3 and by 4.
| 324: 32 × 2 − 4 = 60.
|-
|id=13 rowspan=3| '''[[13 (number)|13]]'''
| Form the [[alternating sum]] of blocks of three from right to left.<ref name="alternating-sum-of-blocks-of-three"/>
| 2,911,272: −2 + 911 − 272 = 637
| 637: 63 - 63 = 0.
|-
|id=14 rowspan=2| '''[[14 (number)|14]]'''
| It is divisible by 2 and by 7.<ref name="product-of-coprimes"/>
| 224: it is divisible by 2 and by 7.
| 364: 3 × 2 + 64 = 70.<br />1764: 17 × 2 + 64 = 98.
|-
|id=15| '''[[15 (number)|15]]'''
| It is divisible by 3 and by 5.<ref name="product-of-coprimes"/>
| 390: it is divisible by 3 and by 5.
|-
|id=16 rowspan=4| '''[[16 (number)|16]]'''
|style="border-bottom: hidden;"| If the thousands digit is even, examine the number formed by the last three digits.
|style="border-bottom: hidden;"| 254,176: 176.
| 157,648: 7,648 = 478 × 16.
|-
|id=17| '''[[17 (number)|17]]'''
| Subtract 5 times the last digit from the rest.
| 221: 22 − 1 × 5 = 17.
|-
|id=18| '''[[18 (number)|18]]'''
| It is divisible by 2 and by 9.<ref name="product-of-coprimes"/>
| 342: it is divisible by 2 and by 9.
|-
|id=19| '''[[19 (number)|19]]'''
| Add twice the last digit to the rest.
| 437: 43 + 7 × 2 = 57.
|-
|id=20 rowspan=2| '''[[20 (number)|20]]'''
| It is divisible by 10, and the tens digit is even.
| 360: is divisible by 10, and 6 is even.