Find the least positive integer that leaves remainders of 1, 2 and 3 when divided by 3, 5 and 7 respectively.
that not even me but ok?
wala akong sagot
To find the least positive integer that satisfies the given conditions, we can use the Chinese Remainder Theorem (CRT). The Chinese Remainder Theorem states that if we have a set of congruences with pairwise relatively prime moduli, then there exists a unique solution modulo the product of the moduli.
In this case, we have three congruences:
x ≡ 1 (mod 3)
x ≡ 2 (mod 5)
x ≡ 3 (mod 7)
First, we find the product of the moduli: 3 × 5 × 7 = 105.
Next, we find the inverses of each modulus with respect to the product. To find the inverse of a modulus, we can use the Extended Euclidean Algorithm.
For mod 3:
105 ÷ 3 = 35 remainder 0
3 ÷ 0 = 0 remainder 3
So, the inverse of mod 3 is 0.
For mod 5:
105 ÷ 5 = 21 remainder 0
5 ÷ 0 = 0 remainder 5
So, the inverse of mod 5 is 0.
For mod 7:
105 ÷ 7 = 15 remainder 0
7 ÷ 0 = 0 remainder 7
So, the inverse of mod 7 is 0.
Now, we calculate the Chinese Remainder Theorem solution using the following equation:
x ≡ (1 × 105 × 0) + (2 × 21 × 0) + (3 × 15 × 0) (mod 105)
Simplifying this equation, we get:
x ≡ 0 + 0 + 0 (mod 105)
x ≡ 0 (mod 105)
Therefore, the least positive integer that satisfies the given conditions is 0.