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?
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.
+118690 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
