Ibraheem's divisibility theory

Ibraheem's divisibility theory is a new way for division with only addition and multiplication operations and only one easy division operation .
Where the known division ways uses a lot of division operations which take more time for computer to calculate than addition and multiplication .
The theory can help us to know if any number divides any prime number except 2,5 which are easy to check for them .

For example we can know if the number m=1718364m=1718364 divide other prime number p=13p=13 with only one division operation ,

If the prime number p units digit is 11 :
We can write it as x1x1, where xx is the whole number without the units digit .
We have to multiply m units digit by (9x+1)(9x+1 ),and add it to (m without the units digit) ,
For example if m=137m=137,and p=11p=11
》 so x=1x=1,
mm without the units digit = 1313,
mm units digit =77
So :p divides m (11divides137)(11 divides 137) only if 1111 divides 7(91+1)+13=837*(9*1+1)+13=83 .

If p units digit is 33 , we have to multiply the units digit of m by 3x+13x+1 and continue with the same way .

If p units digit is 77 , we have to multiply the units digit of m by 7x+57x+5 and continue with the same way .

If p units digit is 99 , we have to multiply the units digit of m by x+1x+1 and continue with the same way .

We can repeat the same function until we reach small numbers , which we can divide by easily ,

Examples :

1)Does 3131 divide 17671767 ?
3131 units digit is 11
So we have to use 9x+19x+1 to multiply by ,
Where x=3x=3, so we have to multiply by 9x+1=93+1=289x+1=9*3+1=28
3131 divides 17671767 only if 3131 divides 176+728=372176+7*28=372
3131 divides 17671767 only if 3131 divides 37+228=9337+2*28=93 .
93/31=393/31=33131 divides 9393》 31) divides 17671767 .

2)Does 4343 divide 5822258222 ?
4343 units digit is 33
So we have to use 3x+13x+1 to multiply by ,
Where x=4x=4, so we have to multiply by 3x+1=34+1=133x+1=3*4+1=13
4343 divides 5822258222 only if 4343 divides 5822+213=58485822+2*13=5848 .
4343 divides 5822258222 only if 4343 divides 584+813=688584+8*13=688 .
4343 divides 5822258222 only if 4343 divides 68+813=17268+8*13=172
4343 divides 5822258222 only if \43) divides (17+213=43(17+2*13=43
43/43=143/43=14343 divides 43434343 divides 5822258222 .

3)Does 77 divide 30733073 ?
77 units digit is 77
So we have to use 7x+57x+5 to multiply by ,
Where x=0x=0, so we have to multiply by 7x+5=70+5=57x+5=7*0+5=5
77 divides 30733073 only if 77 divides 307+35=322307+3*5=322 .
77 divides 30733073 only if 77 divides 32+25=4232+2*5=42 .
42/7=642/7=677 divides \42)》 77 divides 30733073 .

4)Does 379379 divide 136440136440 ?
379379 units digit is 99
So we have to use x+1x+1 to multiply by ,
Where x=37x=37, so we have to multiply by x+1=37+1=38x+1=37+1=38
379379 divides 136440136440 only if 379379 divides 13644+038=1364413644+0*38=13644 .
379379 divides 136440136440 only if 379379 divides 1364+438=15161364+4*38=1516 .
379379 divides 136440136440 only if 379379 divides 151+638=379151+6*38=379 .
379/379=1379/379=1379379 divides 379379379379 divides 136440136440 .

Note by ابراهيم فقرا
7 months, 3 weeks ago

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Great idea.

Max Max - 7 months, 3 weeks ago

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Thank you 😄

ابراهيم فقرا - 7 months, 3 weeks ago

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