Delbarhet
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Aktivitet
Receptomvandling i grupper om 4-5 personer. Om klassen ska tillverka 40 stycken chokladbollar, hur mycket ingredienser behövs för varje grupp och boll. Varje grupp måste redovisa tydliga beräkningar för att hämta ut ingredienser.
Teori
Delbarhet är en matematisk operation
Definition delbarhet:
Ett heltal a är delbart med ett heltal b (b ≠ 0) om a / b = c sådant att kvoten c är ett heltal.
Delbarhet med 2, 3 och 5
När det kommer till delbarheten med våra minsta primtal så ser vi att de sammansatta talen har några gemensamma egenskaper.
Delbarhet med 2:
Alla jämna tal är delbara med 2.
Exempel: 4, 16, 20, 38, 56, 1576
Delbarhet med 3:
Alla tal vars siffersumma är delbar med 3 är delbara med 3.
Exempel: 36, 528, 945, 7521
Delbarhet med 5:
Alla tal där den sista siffran är en 0:a eller en 5:a är delbara med 5.
Exempel: 35, 340, 785, 6345
| Divisor | Divisibility condition | Examples |
|---|---|---|
| 1 | No special condition. Any integer is divisible by 1. | 2 is divisible by 1. |
| 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), [[[:Mall:Google books]] p. 100–101]</ref><ref name="last-m-digits">A number is divisible by 2m, 5m or 10m if and only if the number formed by the last m digits is divisible by that number. See Richmond & Richmond (2009), [[[:Mall:Google books]] p. 105]</ref> | 1294: 4 is even. |
| 3 | Sum the digits. The result must be divisible by 3.<ref name="Pascal's-criterion"/><ref name="apostol-1976">Apostol (1976), [[[:Mall:Google books]] p. 108]</ref><ref name="Richmond-Richmond-2009">Richmond & Richmond (2009), [[[:Mall:Google books]] 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. 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. |
| Subtract the quantity of the digits 2, 5, and 8 in the number from the quantity of the digits 1, 4, and 7 in the number. The result must be 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. | |
| 4 | The last two digits form a number that is divisible by 4.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/> | 40,832: 32 is divisible by 4. |
| If the tens digit is even, the ones digit must be 0, 4, or 8. If the tens digit is odd, the ones digit must be 2 or 6. |
40,832: 3 is odd, and the last digit is 2. | |
| Twice the tens digit, plus the ones digit is divisible by 4. | 40832: 2 × 3 + 2 = 8, which is divisible by 4. | |
| 5 | The last digit is 0 or 5.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/> | 495: the last digit is 5. |
| 6 | It is divisible by 2 and by 3.<ref name="product-of-coprimes">Richmond & Richmond (2009), [[[:Mall:Google books]] Section 3.4 (Divisibility Tests), Theorem 3.4.3, p. 107]</ref> | 1458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6. |
| 7 | Forming an alternating sum of blocks of three from right to left gives a multiple of 7<ref name="Richmond-Richmond-2009"/><ref name="alternating-sum-of-blocks-of-three">Kisačanin (1998), [[[:Mall:Google books]] p. 101]</ref> | 1,369,851: 851 − 369 + 1 = 483 = 7 × 69 |
| Subtracting 2 times the last digit from the rest gives a multiple of 7. (Works because 21 is divisible by 7.) | 483: 48 − (3 × 2) = 42 = 7 × 6. | |
| Adding 5 times the last digit to the rest gives a multiple of 7. (Works because 49 is divisible by 7.) | 483: 48 + (3 × 5) = 63 = 7 × 9. | |
| Adding 3 times the first digit to the next gives a multiple of 7 (This works because 10a + b − 7a = 3a + b − last number has the same remainder) | 483: 4×3 + 8 = '20' remainder 6,
203: 2×3 + 0 = '6' 63: 6×3 + 3 = 21. | |
| Adding the last two digits to twice the rest gives a multiple of 7. (Works because 98 is divisible by 7.) | 483,595: 95 + (2 × 4835) = 9765: 65 + (2 × 97) = 259: 59 + (2 × 2) = 63. | |
| Multiply each digit (from right to left) by the digit in the corresponding position in this pattern (from left to right): 1, 3, 2, -1, -3, -2 (repeating for digits beyond the hundred-thousands place). Then sum the results. | 483,595: (4 × (-2)) + (8 × (-3)) + (3 × (-1)) + (5 × 2) + (9 × 3) + (5 × 1) = 7. | |
| 8 | ||
| If the hundreds digit is odd, the number obtained by the last two digits plus 4 must be divisible by 8. | 352: 52 + 4 = 56. | |
| Add the last digit to twice the rest. The result must be divisible by 8. | 56: (5 × 2) + 6 = 16. | |
| The last three digits are divisible by 8.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/> | 34,152: Examine divisibility of just 152: 19 × 8 | |
| Add four times the hundreds digit to twice the tens digit to the ones digit. The result must be divisible by 8. | 34,152: 4 × 1 + 5 × 2 + 2 = 16 | |
| 9 | Sum the digits. The result must be divisible by 9.<ref name="Pascal's-criterion"/><ref name="apostol-1976"/><ref name="Richmond-Richmond-2009"/> | 2880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9. |
| 10 | The last digit is 0.<ref name="last-m-digits"/> | 130: the last digit is 0. |
| 11 | Form the alternating sum of the digits. The result must be divisible by 11.<ref name="Pascal's-criterion"/><ref name="Richmond-Richmond-2009"/> | 918,082: 9 − 1 + 8 − 0 + 8 − 2 = 22 = 2 × 11. |
| Add the digits in blocks of two from right to left. The result must be divisible by 11.<ref name="Pascal's-criterion"/> | 627: 6 + 27 = 33 = 3 × 11. | |
| Subtract the last digit from the rest. The result must be divisible by 11. | 627: 62 − 7 = 55 = 5 × 11. | |
| Add the last digit to the hundredth place (add 10 times the last digit to the rest). The result must be divisible by 11. | 627: 62 + 70 = 132: 13 + 20 = 33 = 3 × 11. | |
| If the number of digits is even, add the first and subtract the last digit from the rest. The result must be divisible by 11. | 918,082: the number of digits is even (6) → 1808 + 9 − 2 = 1815: 81 + 1 − 5 = 77 = 7 × 11 | |
| If the number of digits is odd, subtract the first and last digit from the rest. The result must be divisible by 11. | 14,179: the number of digits is odd (5) → 417 − 1 − 9 = 407 = 37 × 11 | |
| 12 | It is divisible by 3 and by 4.<ref name="product-of-coprimes"/> | 324: it is divisible by 3 and by 4. |
| Subtract the last digit from twice the rest. The result must be divisible by 12. | 324: 32 × 2 − 4 = 60 = 5 × 12. | |
| 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 |
| Add 4 times the last digit to the rest. The result must be divisible by 13. | 637: 63 + 7 × 4 = 91, 9 + 1 × 4 = 13. | |
| Subtract the last two digits from four times the rest. The result must be divisible by 13. | 923: 9 × 4 - 23 = 13. | |
| Subtract 9 times the last digit from the rest. The result must be divisible by 13. | 637: 63 - 7 × 9 = 0. | |
| 14 | It is divisible by 2 and by 7.<ref name="product-of-coprimes"/> | 224: it is divisible by 2 and by 7. |
| Add the last two digits to twice the rest. The result must be divisible by 14. | 364: 3 × 2 + 64 = 70. 1764: 17 × 2 + 64 = 98. | |
| 15 | It is divisible by 3 and by 5.<ref name="product-of-coprimes"/> | 390: it is divisible by 3 and by 5. |
| 16 | ||
| If the thousands digit is odd, examine the number formed by the last three digits plus 8. | 3408: 408 + 8 = 416. | |
| Add the last two digits to four times the rest. The result must be divisible by 16. | 176: 1 × 4 + 76 = 80.
1168: 11 × 4 + 68 = 112. | |
| Examine the last four digits.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/> | 157,648: 7,648 = 478 × 16. | |
| 17 | Subtract 5 times the last digit from the rest. | 221: 22 − 1 × 5 = 17. |
| Subtract the last two digits from two times the rest. | 4,675: 46 × 2 - 75 = 17. | |
| 18 | It is divisible by 2 and by 9.<ref name="product-of-coprimes"/> | 342: it is divisible by 2 and by 9. |
| 19 | Add twice the last digit to the rest. | 437: 43 + 7 × 2 = 57. |
| Add 4 times the last two digits to the rest. | 6935: 69 + 35 × 4 = 209. | |
| 20 | It is divisible by 10, and the tens digit is even. | 360: is divisible by 10, and 6 is even. |
| The number formed by the last two digits is divisible by 20.<ref name="last-m-digits"/> | 480: 80 is divisible by 20. | |
| 21 | Subtract twice the last digit from the rest. | 168: 16 − 8 × 2 = 0. |
| It is divisible by 3 and by 7.<ref name="product-of-coprimes"/> | 231: it is divisible by 3 and by 7. | |
| 22 | It is divisible by 2 and by 11.<ref name="product-of-coprimes"/> | 352: it is divisible by 2 and by 11. |
| 23 | Add 7 times the last digit to the rest. | 3128: 312 + 8 × 7 = 368. 36 + 8 × 7 = 92. |
| Add 3 times the last two digits to the rest. | 1725: 17 + 25 × 3 = 92. | |
| 24 | It is divisible by 3 and by 8.<ref name="product-of-coprimes"/> | 552: it is divisible by 3 and by 8. |
| 25 | The number formed by the last two digits is divisible by 25.<ref name="last-m-digits"/> | 134,250: 50 is divisible by 25. |
| 26 | It is divisible by 2 and by 13.<ref name="product-of-coprimes"/> | 156: it is divisible by 2 and by 13. |
| 27 | Sum the digits in blocks of three from right to left. The result must be divisible by 27. | 2,644,272: 2 + 644 + 272 = 918. |
| Subtract 8 times the last digit from the rest. The result must be divisible by 27. | 621: 62 − 1 × 8 = 54. | |
| Subtract the last two digits from 8 times the rest. The result must be divisible by 27. | 6507: 65 × 8 - 7 = 520 - 7 = 513 = 27 × 19. | |
| 28 | It is divisible by 4 and by 7.<ref name="product-of-coprimes"/> | 140: it is divisible by 4 and by 7. |
| 29 | Add three times the last digit to the rest. The result must be divisible by 29. | 348: 34 + 8 × 3 = 58. |
| Add 9 times the last two digits to the rest. | 5510: 55 + 10 × 9 = 145 = 5 × 29. | |
| 30 | It is divisible by 3 and by 10.<ref name="product-of-coprimes"/> | 270: it is divisible by 3 and by 10. |