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# Number Theory

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N is a four-digit positive integer. Dividing N by 9, the remainder is 5. Dividing n by 7, the remainder is 2. Dividing n by 5, the remainder is 1. What is the smallest possible value of N?

May 11, 2022

#1
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N mod 9==5

N mod 7==2

N mod 5==1

LCM(9,7,5) ==315

N ==315m  + 86, where m==0, 1, 2, 3......etc.

When m==3, you have:

N==[315 x 3]   +  86 ==1,031 - is the smallest 4-digit integer that satisfies the 3 congruences.

May 11, 2022
#2
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N is a four-digit positive integer. Dividing N by 9, the remainder is 5. Dividing n by 7, the remainder is 2. Dividing n by 5, the remainder is 1. What is the smallest possible value of N?

Thanks guest, it is nice to have the answer

...but I would like to see it done without the omputer.

I have tried to do this  it without a computer program.

Let the number be N where

N=1000a + 100b + 10c + d

since it is 1 (mod5)        d = 1 or 6   (*)

since it is 5 (mod9)        a+b+c+d (mod9) = 5

so

If    d=1    a+b+c = 4 (mod9)   so  a+b+c = 4  or  13  or  22

if    d=6    a+b+c = 8 (mod9)   so   a+b+c = 8 or 17 or  26

So far we have six possibilities for the sum of a,b and c, we want the smallest number so it will probably be one of the smaller ones but that doesn't help much.

Now  N(mod7) = 2

1000a + 100b + 10c + d (mod7) =2

6a     +    2b   +  3c  + d (mod7) =2

If d=1 then  6a+2b+3c = -a+2b+3c = 1 (mod7)

so    -a+2b+3c = 1 or 8 or 15 or 22  or 29  or  36  or 43

If d=6 then  -a+2b+3c = -4 (mod7)

so       -a+2b+3c = -4 or 3 or 10 or 17 or 24 or 31 or 38

summarizing what I have

d =1 or 6

If d=1 then

a+b+c = 4  or  13  or  22

-a+2b+3c = 1 or 8 or 15 or 22  or 29  or  36  or 43

If d=6 then

a+b+c = 8 or 17 or  26

-a+2b+3c = -4 or 3 or 10 or 17 or 24 or 31 or 38

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I want the smallest one so hopefully a will be 1  let me look at that

If d=1 (and a=1) then

b+c = 3  or  12

2b+3c = 2 or 9 or 16 or 23  or 30  or  37  or 44

If d=6 (and a=1) then

b+c = 7 or 16 or  25

2b+3c = -3 or 4 or 11 or 18 or 25 or 32 or 39

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If d=1 (and a=1) then

2b+2c+c = 2 or 9 or 16 or 23  or 30  or  37  or 44

b+c = 3  or  12

2b+2c = 6 or 24

if

If 2b+2c=6 then c=3    and we have the number   1031

If d=6 (and a=1) then

2b+2c +c = -3 or 4 or 11 or 18 or 25 or 32 or 39

b+c = 7 or 16 or  25

2b+2c = 14 or 32  (50 is too big)

If

2b+2c = 14 then c=4  and so b=3   so the number is  1346

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So

1031 is the smallest one  (just as guest already determined with his/her computer program.

That was an exteme effort!    LOL

May 11, 2022
edited by Melody  May 12, 2022