When a number is divided by 7 and 9 , the remainder is 4 and 3. what will be the remainder of a same no. when it is divided by 37.
To find the remainder when a number is divided by 37, we can use the Chinese remainder theorem.
Since the remainder when the number is divided by 7 is 4, we can express it as:
Number ≡ 4 (mod 7)
Similarly, since the remainder when the number is divided by 9 is 3, we can express it as:
Number ≡ 3 (mod 9)
Now, we need to solve these two congruences simultaneously to find a common solution.
Using the Chinese remainder theorem, we can find a number that satisfies both congruences.
To find the solution, multiply the first congruence (mod 7) by 9 and the second congruence (mod 9) by 7. This gives us:
9 * Number ≡ 9 * 4 (mod 7)
7 * Number ≡ 7 * 3 (mod 9)
Simplifying these congruences, we get:
9 * Number ≡ 36 (mod 7)
7 * Number ≡ 21 (mod 9)
The first congruence can be simplified as:
2 * Number ≡ 1 (mod 7)
And the second congruence becomes:
7 * Number ≡ 3 (mod 9)
Now, we need to find the inverse of 2 modulo 7. Since 2 and 7 are coprime, their inverse exists. We can find it by trial and error:
2 * 4 ≡ 1 (mod 7)
So, the inverse of 2 modulo 7 is 4. Multiplying the congruence by 4, we get:
Number ≡ 4 (mod 7)
Next, we find the inverse of 7 modulo 9. Once again, by trial and error:
7 * 4 ≡ 1 (mod 9)
Hence, the inverse of 7 modulo 9 is 4. Multiplying the congruence by 4, we get:
Number ≡ 12 (mod 9)
Number ≡ 3 (mod 9)
Now we have two congruences:
Number ≡ 4 (mod 7)
Number ≡ 3 (mod 9)
Using the Chinese remainder theorem, we can combine these two congruences to find the solution modulo 63 (7 * 9):
Number ≡ (4 * 9 * 4 + 3 * 7 * 3) (mod 63)
Number ≡ 144 + 63 (mod 63)
Number ≡ 207 (mod 63)
Finally, to find the remainder when 207 is divided by 37, we can divide 207 by 37, which gives us:
207 = 37 * 5 + 12
Therefore, the remainder when the number is divided by 37 is 12.