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# number theory

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How many positive integers less than or equal to 6*7*8*9 solve the system of congruences:

m = 5 (mod 6)

m = 4 (mod 7)

m = 3 (mod 8)

m = 2 (mod 9)

m = 1 (mod 10)

Feb 18, 2022

#1
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m = 5 (mod 6)

m = 4 (mod 7)

m = 3 (mod 8)

m = 2 (mod 9)

m = 1 (mod 10)

LCM[6, 7, 8, 9, 10] ==2,520

The smallest integers that will satisfy all five congruences < 3,024  are:

11  and  2,521 (2,520 + 1)

Feb 18, 2022
#2
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Guest, you made a small mistake.  I like seeing your answers though. :)

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I'm going to try and solve in pairs

m=-1 mod 6 and 1 mod 10

11  works

m=4 mod 7 and 1 mod 10

11 works

now I look, 11 works for all of them

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 Mod number Prime factors 6 2 3 7 7 8 2*2*2 9 3*3 10 2 5 prime factors needed 2*2*2=8 3*3=9 5 7 5*7*8*9

So the solutions will be of the form    m=(5*7*8*9)k + 11

the first 2 are 11 and  (5*7*8*9) + 11

the 3rd one is         2(5*7*8*9) + 11   and this is bigger than    6*7*8*9

so

There are just 2          they are     11 and  2531

Feb 19, 2022