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When S is divided by 5 remainder is 3, when it is divided by 7, remainder is 4. If K+S is divisible by 35, what is the least possible value of K?

Or by Chinese modulus theorem, we have, from what is provided,
S=35n +18 ( n is integer)
S+ K= 35n +18+ K, 35n is divisible by 35 ---> the least possible value of K= 17 so that S+ K= 35n + 35

When S is divided by 5 remainder is 3, when it is divided by 7, remainder is 4. If K+S is divisible by 35, what is the least possible value of K?

Or by Chinese modulus theorem, we have, from what is provided, S=35n +18 ( n is integer) S+ K= 35n +18+ K, 35n is divisible by 35 ---> the least possible value of K= 17 so that S+ K= 35n + 35

When S is divided by 5 remainder is 3, when it is divided by 7, remainder is 4. If K+S is divisible by 35, what is the least possible value of K?

Or by Chinese modulus theorem, we have, from what is provided, S=35n +18 ( n is integer) S+ K= 35n +18+ K, 35n is divisible by 35 ---> the least possible value of K= 17 so that S+ K= 35n + 35

Pls explain how you got the below stmt

S=35n +18 ( n is integer)

We have:
S= 3(mod 5) or S= 4(mod 7)
S= 5m+ 3 ( m is integer) --> 5m+3= 4 ( mod 7) --> 5m= 1(mod 7) --> m= 3 ( mod 7 ) (the method is to try different remainders of 7, starting from 1) --> m= 7n + 3 ( n is integer) --> substitute m by n, we have:
S= 35n+ 18

The above method is so-called Chinese modulus theorem . I saw some above post have the link to this theorem.

When S is divided by 5 remainder is 3, when it is divided by 7, remainder is 4. If K+S is divisible by 35, what is the least possible value of K?

Or by Chinese modulus theorem, we have, from what is provided, S=35n +18 ( n is integer) S+ K= 35n +18+ K, 35n is divisible by 35 ---> the least possible value of K= 17 so that S+ K= 35n + 35

Pls explain how you got the below stmt

S=35n +18 ( n is integer)

We have: S= 3(mod 5) or S= 4(mod 7) S= 5m+ 3 ( m is integer) --> 5m+3= 4 ( mod 7) --> 5m= 1(mod 7) --> m= 3 ( mod 7 ) (the method is to try different remainders of 7, starting from 1) --> m= 7n + 3 ( n is integer) --> substitute m by n, we have: S= 35n+ 18

The above method is so-called Chinese modulus theorem . I saw some above post have the link to this theorem.

I like to add that the CRT works only if the mod values are relatively prime.

For three facts relating to S, I think it can be worked out but would require a very exhaustive method that invovles euclidean algorithm...

Here's a link, that shows how to solve for 3 facts using the CRT.