A number on being divided by 10 and 11 successively leaves the remainders 5 and 7. Find the number.
24
Hmmm.
24 = 2*10 + 4
24 = 2*11 + 2
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To find the number, we can solve this problem using the Chinese Remainder Theorem.
Let's assume the unknown number is represented by 'x'.
According to the problem, when x is divided by 10, the remainder is 5. This can be written as:
x ≡ 5 mod 10 (1)
Similarly, when x is divided by 11, the remainder is 7. This can be written as:
x ≡ 7 mod 11 (2)
To solve this system of congruences, we can use the Chinese Remainder Theorem (CRT).
The first step is to find the modular inverses for each modulus. In this case, we need to find the modular inverses of 10 and 11.
To find the modular inverse of 10 (mod 11), we need to find a number 'a' such that:
10 * a ≡ 1 mod 11
By checking the possible values for 'a', we find that a = 10 satisfies this equation. Therefore, the modular inverse of 10 (mod 11) is 10.
Similarly, to find the modular inverse of 11 (mod 10), we need to find a number 'b' such that:
11 * b ≡ 1 mod 10
By checking the possible values for 'b', we find that b = 9 satisfies this equation. Therefore, the modular inverse of 11 (mod 10) is 9.
Now that we have the modular inverses, we can apply the Chinese Remainder Theorem to find the value of 'x'.
Using the formula:
x ≡ (a * (remainder_1) * (modular_inverse_1) + b * (remainder_2) * (modular_inverse_2)) mod (modulus_1 * modulus_2)
Substituting the given values, we have:
x ≡ (10 * 5 * 10 + 9 * 7 * 9) mod (10 * 11)
x ≡ (500 + 567) mod 110
x ≡ 1067 mod 110
To find the smallest positive value of 'x', we can subtract multiples of 110 until we get a result less than 110:
x = 1067 - 110 * 9
x = 1067 - 990
x = 77
Therefore, the number that satisfies the given conditions is 77.