Tool for discriminating multiple of 7

Abstract

To solve the problem in which: discriminating whether a certain natural number (10a+b) is a multiple of 7 often requires time and effort for calculation; the present invention allows relatively easy discrimination as to whether two- or more-digit natural number (10a+b) is a multiple of 7; the invention can be useful for business processing, information processing, and calculation processing, utilization and application for discrimination of a multiple of 7, a prime number, as well as factorization.SOLUTION: In discriminating whether a certain natural number (10a+b) is a multiple of 7, if 10a+b=7N (a is a one- or more-digit natural number, b is 0 or a one-digit natural number, N is a natural number of 1 or more, and so on) holds up, (a+5b)=7(5 N-7a) is true. Performing such an operation on the original natural number (10a+b) reduces the number of digits of the original natural number basically by one in a single operation, and sequentially performing such an operation on a number obtained by the above-described operation ultimately results in 7 or 49 (multiple of 7), which reveals that the original natural number is a multiple of 7. This allows provision of a tool for discriminating a multiple of 7 by using repetition of the above-described operation on the original natural number.

Description

[Technical field]

[0001] The present invention relates to tools such as cards, documents, books, emails, personal computers, smartphones, mobile phones, electronic media, magnetic media, recording media, printed matter, displays, calculation materials, educational materials, office equipment, calculators, electronic devices, electrical equipment, videos, images, speeches, lectures, information, electronic information, broadcasts or broadcasts that are useful for factorizing complex numbers or numbers with many digits (certain natural numbers), finding their divisors, identifying prime numbers, and counting processing, and for determining whether a number is a multiple of 7, and further for use and application in counting processing and information processing. [Background technology]

[0002] It is common to factor or reduce large numbers (certain natural numbers), and it has long been known to determine prime numbers such as 2, 3, 4, 5, 6, 8, 9, 11, etc., and multiples of numbers that are several times larger than these prime numbers.

[0003] Furthermore, determining multiples of 7 is described in the following patent documents: [Prior art documents] [Patent documents]

[0004] [Patent Document 1] Japanese Patent Application Publication No. 06-149149 [Patent Document 2] Japanese Patent Application Publication No. 07-005802 [Patent Document 3] Japanese Patent Application Publication No. 07-005803 [Patent Document 4] Japanese Patent Application Publication No. 07-013481 [Patent Document 5] JP 2023-026269 A Summary of the Invention [Problem to be solved by the invention]

[0005] The present invention makes it easier to determine multiples of seven that are two or more digits long than the prior art documents, and is particularly useful for determining whether natural numbers with many digits are multiples of seven, and is useful for clerical work, information processing, and counting, as well as for other applications such as determining prime numbers and factoring. Although it was possible to distinguish multiples of 7 in conventional prior art documents, the challenge was to make it even easier to distinguish them. [Means for solving the problem]

[0006] The present invention uses natural numbers that are multiples of 7, represented as 7N (N is a natural number equal to or greater than 1, the same applies below). If we express this as (10a+b) (where a is a natural number with one or more digits, b is a natural number that is 0 or has one digit, and the same applies below), the following operation gives us (a+5b)=7(5N-7a), and by performing this operation, we determine that the original natural number (10a+b) is a multiple of 7. Therefore, when we perform this operation (a+5b) on the original natural number (10a+b), the number of digits in the original natural number will basically be reduced by one with each operation, so if we repeat this operation one after another on the number obtained by the above operation, we will eventually become 7 or 49 (a multiple of 7).

[0007] The present invention utilizes and applies the above method to tools for identifying multiples of 7 of two or more digits, such as cards, documents, books, emails, personal computers, mobile phones, smartphones, magnetic media, recording media, electronic media, printed matter, displays, calculation materials, educational materials, office equipment, calculators, electronic devices, electrical equipment, videos, images, lectures, lectures, information, electronic information, broadcasting or broadcasting, etc., and there are no particular limitations on the materials used, and known materials can be used, and the size can also be freely set depending on the purpose of use.

[0008] When the number of digits in a certain natural number (10a+b) becomes large, the above operation (a+5b) requires more and more operations as the number of digits increases. However, by repeating the operation, the number of digits in the natural number being examined basically decreases by one digit each time, so that the final number becomes 7 or 49. Furthermore, even if you perform the operation (a+5b) on 49, it will still be 49 and the number will not change. Other two-digit multiples of seven may require more manipulations, but they all end up as 7. For natural numbers with three or more digits, the number of operations can be increased depending on the number of digits, eventually resulting in a two-digit number that is a multiple of 7, and ultimately becoming 7 or 49. Furthermore, if the original natural number is not a multiple of 7, even if the operation (a+5b) of the present invention is repeated, the final number will not be a multiple of 7.

[0009] The present invention will be specifically described below with reference to examples. The present invention can be used for tools that handle natural numbers with two or more digits (10a+b) (regardless of the number of digits, with no particular upper limit). In the present invention, as an example, numbers with 2 to 10 digits are used. Example 1

[0010] Examples of 2-digit numbers that are multiples of 7 1) About 14 Since the a in the 10th place or higher is 1 and the b in the 1st place is 4, The operation (a+5b)=1+5×4=21. For the 21 obtained here, the a in the 10th place or higher is 2, and the b in the 1st place is 1, so a+5b=2+5×1=7. 2) About 21 Since the a in the 10th place or higher is 2 and the b in the 1st place is 1, a+5b=2+5×1=7. 3) About 28 The number of a's in the 10th place and above is 2, and the number of b's in the 1st place is 8, so a+5b=2+5×8=42. For the 42 obtained here, the a in the 10th place or higher is 4, and the b in the 1st place is 2, so a+5b=4+5×2=14. Please refer to 1) for details on this 14. 4) About 35 The number of a's in the 10th place and above is 3, and the number of b's in the 1st place is 5, so a+5b=3+5×5=28. For details on this 28, please refer to 3). 5) Regarding 42, please refer to 3). 6) About 49 The number of a's in the 10th place and above is 4, and the number of b's in the 1st place is 9, so a+5b=4+5×9=49, When you add a+5b, the number 49 remains unchanged. 7) About 56 Since the a in the 10th place or higher is 5 and the b in the 1st place is 6, a+5b=5+5×6=35. For details on this 35, please refer to 4). 8)About 63 The number of a's in the 10th place and above is 6, and the number of b's in the 1st place is 3, so a+5b=6+5×3=21. For the 21 obtained here, the a in the 10th place or higher is 2, and the b in the 1st place is 1, so a+5b=2+5×1=7. 7) About 70 Since the a in the 10th place or higher is 7 and the b in the 1st place is 0, a+5b=7+5×0=7. 8) About 77 The a in the 10th place and above is 7, and the b in the 1st place is 7, so a+5b=7+5×7=42. The 42 obtained here are shown in 3), so please refer to them. 9) About 84 The number of a's in the 10th place and above is 8, and the number of b's in the 1st place is 4, so a+5b=8+5×4=28 The 28 results obtained here are shown in 3), so please refer to them. 10)About 91 The a in the 10th place or higher is 9, and the b in the 1st place is 1, so a+5b=9+5×1=14 The 14 obtained here is shown in 1), so please refer to it. 11)About 98 The a in the 10th place or higher is 9, and the b in the 1st place is 1, so a+5b=9+5×8=49. Example 2

[0011] Examples of 3-digit numbers that are multiples of 7 1) About 259 Because a, which is 10th place or higher, is 25, and b, which is 1st place, is 9. a+5b=25+5×9=70. For the 70 obtained here, the a in the 10th place and above is 7, and the b in the 1st place is 0, so a+5b=7+5×0=7. 2) About 427 The number of a's in the 10th place and above is 42, and the number of b's in the 1st place is 7, so a+5b=42+5×7=77. For the 77 obtained here, the a in the 10th place and above is 7, and the b in the 1st place is 7, so a+5b=7+5×7=42, where 42 is a multiple of 7. 3) About 973 The number of a's in the 10th place and above is 97, and the number of b's in the 1st place is 3, so a+5b=97+5×3=112. For the 112 obtained here, the a in the 10th place and above is 11, and the b in the 1st place is 2, so a+5b=11+5×2=21, where 21 is a multiple of 7. Example 3

[0012] Examples of 4-digit numbers that are multiples of 7 Determine whether a four-digit natural number, such as 2191, is a multiple of 7 or not. 1) About 2191 The number of a's in the 10th place and above is 219, and the number of b's in the 1st place is 1, so a+5b=219+5×1=224. For the 224 obtained here, the a in the 10th place and above is 22, and the b in the 1st place is 4, so a+5b=22+5×4=42, where 42 is a multiple of 7. 2) About 3899 The number a, which is 10th place or higher, is 389, and the number b, which is 1st place, is 9. a+5b=389+5×9=434. For the 434 obtained here, the 10th place and higher a is 43, and the 1st place b is 4, so a+5b=43+5×4=63, where 63 is a multiple of 7. 3) About 9527 The number of a's in the 10th place and above is 952, and the number of b's in the 1st place is 7, so a+5b=952+5×7=987. For the 987 obtained here, the 10th place and above a is 98, and the 1st place b is 7, so a+5b=98+5×7=133. For the 133 obtained here, the 10th place and higher a is 13, and the 1st place b is 3, so a+5b=13+5×3=28, where 28 is a multiple of 7. Example 4

[0013] Examples of 5-digit numbers that are multiples of 7 Determine whether a five-digit natural number, such as 16723, is a multiple of 7 or not. 1) About 16723 The number of a's (10th place and above) is 1672, and the number of b's (1st place) is 3, so a+5b=1672+5×3=1687. Of the 1687 obtained here, the a in the 10th place or higher is 168, and the b in the 1st place is 7, so a+5b=168+5×7=203. For the 203 obtained here, the 10th place and above a is 20 and b is 3, so a+5b=20+5×3=35, where 35 is a multiple of 7. 2) About 24577 The 10th place or higher a is 2457, and the 1st place b is 7, so a+5b=2457+5×7=2492. Of the 2492 obtained here, the a in the 10th place or higher is 249, and the b in 1st place is 2, so a+5b=249+5×2=259. For the 259 obtained here, the 10th place and above a is 25, and the 1st place b is 9, so a+5b=25+5×9=70, where 70 is a multiple of 7. 3) About 65779 The 10th place or higher a is 6577, and the 1st place b is 9, so a+5b=6577+5×9=6622. For the 6622 obtained here, the 10th and higher ranks of a and b are 662 and 2, respectively, so a+5b=662+5×2=672. For the 672 obtained here, the 10th place and above a is 67, and the 1st place b is 2, so a+5b=67+5×2=77, where 77 is a multiple of 7. Example 5

[0014] Examples of 6-digit numbers that are multiples of 7 Determine whether the six-digit natural number 564571 is a multiple of 7. 1) About 564571 The 10th place or higher of a is 56457, and the 1st place of b is 1, so a+5b=56457+5×1=56462. For the 56462 obtained here, the 10th place or higher a is 5646, and the 1st place b is 2, so a+5b=5646+5×2=5656. Of the 5656 obtained here, the a in the 10th place or higher is 565, and the b in 1st place is 6, so a+5b=565+5×6=595. For the 595 obtained here, the 10th place and higher a is 59, and the 1st place b is 5, so a+5b=59+5×5=84. For the 84 obtained here, the 10th place and above a is 8, and the 1st place b is 4, so a+5b=8+5×4=28, where 28 is a multiple of 7. Example 6

[0015] Examples of 7-digit numbers that are multiples of 7 Determine whether the seven-digit natural number 2539341 is a multiple of 7. Here, the a in (10a+b) is 253934 and the b is 1. About this 2539341 a+5b=253934+5×1=253939 For the 253939 obtained here, the 10th place or higher a is 25393, and the 1st place b is 9, so a+5b=25393+5×9=25438. For the 25438 obtained here, the 10th place or higher a is 2543, and the 1st place b is 8, so a+5b=2543+5×8=2583. For the 2583 obtained here, the 10th place and higher a is 258, and the 1st place b is 3, so a+5b=258+5×3=273. For the 273 obtained here, the 10th place and above a is 27, and the 1st place b is 3, so a+5b=27+5×3=42, where 42 is a multiple of 7. Example 7

[0016] Examples of 8-digit numbers that are multiples of 7 Determine whether the 8-digit natural number 90793199 is a multiple of 7. Here, the a in (10a+b) is 9079319 and the b is 9. a+5b=9079319+5×9=9079364. For the 9079364 obtained here, the 10th place or higher a is 907936, and the 1st place b is 4, so a+5b=907936+5×4=907956. For the 907956 obtained here, the 10th place or higher a is 90795, and the 1st place b is 6, so a+5b=90795+5×6=90825. For the 90825 obtained here, the 10th place or higher a is 9082, and the 1st place b is 5, so a+5b=9082+5×5=9107. For the 9107 obtained here, the 10th place and higher a is 910, and the 1st place b is 7, so a+5b=910+5×7=945. For the 945 obtained here, the 10th place and higher a is 94, and the 1st place b is 5, so a+5b=94+5×5=119. For the 119 obtained here, the 10th place and higher a is 11, and the 1st place b is 9, so a+5b=11+5×9=56, where 56 is a multiple of 7. Example 8

[0017] Examples of 10-digit numbers that are multiples of 7 Determine whether the 10-digit natural number 9753553579 is a multiple of 7. Here, the a in (10a+b) is 975355357 and the b is 9. If we do the operation (a+5b) on this 9753553579, a+5b=975355357+5×9=975355402. Regarding the 975355402 obtained here The 10th place or higher of a is 97535540, and the 1st place of b is 2, so a+5b=97535540+5×2=97535550. Regarding the 97535550 obtained here The 10th place or higher a is 9753555, and the 1st place b is 0, so a+5b=9753555+5×0=9753555. Regarding the 9753555 obtained here The 10th place or higher a is 975355, and the 1st place b is 5, so a+5b=975355+5×5=975380. Regarding the 975,380 obtained here The 10th place or higher of a is 97538, and the 1st place of b is 0, so a+5b=97538+5×0=97538. Regarding the 97538 obtained here The 10th place or higher of a is 9753, and the 1st place of b is 8, so a+5b=9753+5×8=9793. Regarding the 9793 obtained here The number of a's (10th place and above) is 979, and the number of b's (1st place) is 3, so a+5b=979+5×3=994. Regarding the 994 obtained here The 10th place or higher of a is 99, and the 1st place of b is 4, so a+5b=99+5×4=119. Regarding the 119 obtained here The number of a's in the 10th place and above is 11, and the number of b's in the 1st place is 9, so a+5b=11+5×9=56. This 56 is a multiple of 7. Example 1

[0018] Determine whether the five-digit natural number 54321 is a multiple of 7. For 54321, the a in (10a+b) is 5432 and b is 1. If we apply the operation (a+5b) to 54321, we get 5432+5×1=5427. Doing the same operation on 5437 gives 543+5×7==578. Doing the same with 578 gives us 57 + 5 × 8 = 97. If we do the same for 97, we get 9+5×7=44, which is not a multiple of 7. Therefore, we can see that 54321 is not a multiple of 7. Reference Example 2

[0019] Determine whether the five-digit natural number 99999 is a multiple of 7. For 99999, the a in (10a+b) is 9999 and the b is 9, so If you do the operation (a+5b) The result is 9999+5×9=10044. Regarding the 10044 obtained here, The 10th place or higher of a is 1004, and the 1st place of b is 4, so a+5b=1004+5×4=1024. Regarding the 1024 obtained here, The number of a's in the 10th place and above is 102, and the number of b's in the 1st place is 4, so a+5b=102+5×4=122. Regarding the 122 obtained here The number of a's in the 10th place and above is 12, and the number of b's in the 1st place is 2, so a+5b=12+5×2=22. Since 22 is not a multiple of 7, the original natural number 99999 is also not a multiple of 7.

[0020] Determine whether the 8-digit natural number 12345678 is a multiple of 7. For 12345678, the a in (10a+b) is 1234567 and the b is 8. If we do the operation (a+5b) on this 12345678, The result is 1234567+5×8=1234607. Regarding the 1234607 obtained here The 10th place or higher a is 123460, and the 1st place b is 7, so a+5b=123460+5×7=123495. Regarding the 123495 obtained here The 10th place or higher of a is 12349, and the 1st place of b is 5, so a+5b=12349+5×5=12374. Regarding the 12374 obtained here The 10th place or higher of a is 1237, and the 1st place of b is 4, so a+5b=1237+5×4=1257. Regarding the 1257 obtained here The 10th place or higher of a is 125, and the 1st place of b is 7, so a+5b=125+5×7=160. Regarding the 160 obtained here, The a in the 10th place and above is 16, and the b in the 1st place is 0, so a+5b=16+5×0=16. Since 16 is not a multiple of 7, the original natural number 12345678 is It is clear that the number is not a multiple of 7.

[0021] As described above, the present invention makes it easier to determine whether a natural number is a multiple of 7 than ever before, and is not only useful for determining whether a natural number is a prime number, but can also be used and applied to factorization, counting processing, and office work. The present invention can be used not only in education and computer fields that handle counting, but also in a wide range of applications such as cards, documents, books, emails, personal computers, smartphones, mobile phones, magnetic media, recording media, electronic media, printed matter, displays, calculation materials, educational materials, office equipment, calculators, electronic devices, electrical equipment, videos, images, lectures, lectures, information, electronic information, broadcasting or broadcasting, and other industrial fields that determine multiples of 7.

Claims

1. When determining whether a natural number of two or more digits (10a + b) is a multiple of 7, if 10a + b = 7N (where a is a natural number of one or more digits, b is 0 or a single digit natural number, N is a natural number of 1 or more, same below), then (a + 5b) = 7 (5N - 7a). Therefore, if you repeat this operation (a + 5b) on the number obtained by the operation (a + 5b) in order, the number of digits in the original natural number will basically decrease by one digit with each operation, and will eventually become 7 or 49 (a multiple of 7), and it will be determined that the original natural number is a multiple of 7. Here is a tool to determine multiples of 7 by repeating the operation (a + 5b) for the original natural numbers.

2. When determining whether a natural number of two or more digits (10a + b) is a multiple of 7, if 10a + b = 7N (where a is a natural number of one or more digits, b is 0 or a single digit natural number, N is a natural number of 1 or more, same below), then (a + 5b) = 7 (5N - 7a). Therefore, if you repeat this operation (a + 5b) on the number obtained by the operation (a + 5b) in order, the number of digits in the original natural number will basically decrease by one digit with each operation, and will eventually become 7 or 49 (a multiple of 7), and it will be determined that the original natural number is a multiple of 7. Therefore, tools such as cards, documents, books, emails, computers, smartphones, mobile phones, magnetic media, recording media, electronic media, electronic information, printed matter, displays, calculation materials, educational materials, office equipment, calculators, electronic devices, electrical equipment, videos or images that can determine multiples of 7 by repeating the operation (a + 5b) for the original natural numbers.

3. When determining whether a natural number of two or more digits (10a + b) is a multiple of 7, if 10a + b = 7N (where a is a natural number of one or more digits, b is 0 or a single digit natural number, N is a natural number of 1 or more, same below), then (a + 5b) = 7 (5N - 7a). Therefore, if you repeat this operation (a + 5b) on the number obtained by the operation (a + 5b) in order, the number of digits in the original natural number will basically decrease by one digit with each operation, and will eventually become 7 or 49 (a multiple of 7), and it will be determined that the original natural number is a multiple of 7. This is a tool for lectures, speeches, information, broadcasts, etc. that determines multiples of 7 by repeating the operation (a + 5b) for the original natural numbers.