N is a four-digit positive integer. Dividing N by 9 , the remainder is 5. Dividing N by 7, the remainder is 5. Dividing N by 5, the remainder is 1. What is the smallest possible value of N?
i=0;j=0;m=0;t=0;a=(9, 7, 5);r= (5,5,1);c=lcm(a); d=c / a[i];n=d % a[i] ;loop1:m++; if(n*m % a[i] ==1, goto loop, goto loop1);loop:s=(c/a[i]*r[j]*m);i++;j++;t=t+s;m=0;if(i< count a, goto4,m=m);printc,"m + ",t % c;return
OUTPUT: 315 m + 131, where m=0, 1, 2, 3........etc.
When m = 0, the smallest vlaue of N == 131
N is a four-digit positive integer. Dividing N by 9 , the remainder is 5. Dividing N by 7, the remainder is 5. Dividing N by 5, the remainder is 1. What is the smallest possible value of N?
\(1000 \le N \le9999\)
All the pronumerals I use here are positive integers.
N=9k+5
N=7m+5
so
N=63T+5
(N-5)/63=T
\((1000-5)/63=15.7\\ (9999-5)/63=158.6\\\)
So T must be between 16 and 158
N=5g+1
So
the last digit of N must be 1 or 6
N=63T+5
N - 5=63T
***1-5=***6 ***6-5=***1
So the last digit of 63T must also be 6 or 1
3 times what single digit ends in 6 or 1? 3*2=6, 3*7=21
So T must end in a 2 or a 7
So T is a number between 16 and 158 that ends in 2 or 7.
We want the smallest one.
Try T=22
Edit:
Silly me, T=17 is the smallest that fits these restrictions and it will give N=63*17+5=1076.
Just as our guest has already determined with his/her computer program.
N=63*22+5 = 1391
does this really work?
check:
1391 does equal 1 mod 5
1391 does equal 5 mod 9
1391 does equal 5 mod 7
So I believe the answer to be N=1391