This problem from China is almost 2000 years old: Find a number that when divided by 3 gives a remainder of 2, when divided by 5 gives a remainder of 3, and when divided by 7 gives a remainder of 4.
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To solve this problem, we can use the Chinese Remainder Theorem. The first step is to list down the given remainders and corresponding divisors:
Divisor 1: 3
Remainder 1: 2
Divisor 2: 5
Remainder 2: 3
Divisor 3: 7
Remainder 3: 4
We can represent this system of congruences using variables:
N ≡ 2 (mod 3)
N ≡ 3 (mod 5)
N ≡ 4 (mod 7)
Now, we can find a solution by using the Chinese Remainder Theorem. Let's start by finding a number congruent to 1 modulo 3 and divisible by 5 and 7. One such number is 70. Now, we need to find the integer k such that:
70k ≡ 1 (mod 3)
To simplify this equation, we can rewrite it as:
2k ≡ 1 (mod 3)
After some trial and error, we find that k = 2 is a solution since 2 * 2 ≡ 4 ≡ 1 (mod 3). Therefore, the integer k = 2 satisfies the equation.
Now, we can calculate the value of N using the formula:
N = a1 * m1 * b1 + a2 * m2 * b2 + a3 * m3 * b3
where:
a1, a2, a3 are the given remainders (2, 3, 4)
m1, m2, m3 are the product of all divisors except the current divisor (5 * 7, 3 * 7, 3 * 5)
b1, b2, b3 are the solutions to the equation b * m ≡ 1 (mod divisor)
Calculating each component:
N = 2 * (5 * 7) * 2 + 3 * (3 * 7) * 3 + 4 * (3 * 5) * 1
N = 2 * 70 + 3 * 21 + 4 * 15
N = 140 + 63 + 60
N = 263
Therefore, the number we are looking for is 263.
To solve this problem, we can use the Chinese Remainder Theorem, which allows us to find a unique solution to a system of congruences.
Let's break down the given information into the following congruences:
x ≡ 2 (mod 3)
x ≡ 3 (mod 5)
x ≡ 4 (mod 7)
To find a number that satisfies all three congruences, we can follow these steps:
Step 1: Find the product of the moduli. In this case, 3 * 5 * 7 = 105.
Step 2: For each congruence, divide the product of the moduli by the modulus itself to get a new number.:
For x ≡ 2 (mod 3), this new number is 35 (105 / 3).
For x ≡ 3 (mod 5), this new number is 21 (105 / 5).
For x ≡ 4 (mod 7), this new number is 15 (105 / 7).
Step 3: Find the modular inverse of the new number with respect to each modulus. The modular inverse of a number a (mod m) is a number x such that (a * x) ≡ 1 (mod m).
For 35 (mod 3), the modular inverse is 2 (since 35 * 2 ≡ 1 (mod 3)).
For 21 (mod 5), the modular inverse is 1 (since 21 * 1 ≡ 1 (mod 5)).
For 15 (mod 7), the modular inverse is 1 (since 15 * 1 ≡ 1 (mod 7)).
Step 4: Multiply each congruence by the new number and its modular inverse. Then add them together.
For the first congruence:
(2 * 35 * 2) ≡ 2 * 1 * 1 ≡ 7 (mod 3)
For the second congruence:
(3 * 21 * 1) ≡ 6 * 1 * 1 ≡ 6 (mod 5)
For the third congruence:
(4 * 15 * 1) ≡ 4 * 1 * 1 ≡ 4 (mod 7)
Step 5: Add up the results from Step 4.
7 + 6 + 4 = 17
Step 6: Take the result from Step 5 modulo the product of the moduli.
17 (mod 105) ≡ 17
So the number we are looking for is 17.