N is an integer such that N≡179(mod233) and N≡233(mod179).
Determine a three digit positive integer M such that N≡M(mod179×233).
Details and assumptions
You may use the fact that 179 and 233 are primes.
412
how did you do it?
N = 179(mod 233)
N = 233(mod 179)
N = 179 + 233t
179 + 233t = 233(mod 179)
233t = 233 (mod 179)
t = 1
N = M = 179 + 233(1) = 412 (mod 179*233).
108 is 36% of what number? Write and solve a proportion to solve the problem.
108 is 36% of what number? Write and solve a proportion to solve the problem.
108 is 36% of what number? Write and solve a proportion to solve the problem.
108 is 36% of what number? Write and solve a proportion to solve the problem.
To determine a three-digit positive integer M such that N≡M(mod 179×233), we need to solve the system of congruences given.
First, let's break it down.
From the given information, we have:
N ≡ 179 (mod 233) -- Equation (1)
N ≡ 233 (mod 179) -- Equation (2)
To find a solution to this system, we can use the Chinese Remainder Theorem (CRT).
The Chinese Remainder Theorem states that if we have a system of congruences:
x ≡ a (mod m)
x ≡ b (mod n)
where m and n are relatively prime (i.e., they have no common factors), then there exists a unique solution modulo mn, which can be found using the formula:
x ≡ (a * n * n_inv + b * m * m_inv) (mod mn)
Here, n_inv and m_inv represent the modular inverses of n and m, respectively.
Now, let's apply the Chinese Remainder Theorem to equations (1) and (2).
Since 233 and 179 are prime numbers, they are relatively prime. So, we can apply the CRT formula:
N ≡ (179 * 233 * n_inv + 233 * 179 * m_inv) (mod (179 * 233))
To find the modular inverse of a prime number, we can use Fermat's Little Theorem.
For p being a prime number and a being any positive integer less than p:
a^(p-1) ≡ 1 (mod p)
Using this theorem, we can find the modular inverses of 233 and 179.
For 233:
233^(233-1) ≡ 1 (mod 233)
233^(232) ≡ 1 (mod 233)
Therefore, n_inv = 233^(232) ≡ 1 (mod 233)
For 179:
179^(179-1) ≡ 1 (mod 179)
179^(178) ≡ 1 (mod 179)
Therefore, m_inv = 179^(178) ≡ 1 (mod 179)
Substituting these modular inverse values into the CRT formula, we get:
N ≡ (179 * 233 * 1 + 233 * 179 * 1) (mod (179 * 233))
N ≡ 41507 (mod 41607)
Now, to find a three-digit positive integer M, we need to find an equivalent residue of 41507 modulo 41607. Since the question specifies that M should be a three-digit positive integer, we need to subtract or add multiples of 41607 until we get a three-digit number.
41507 - 41607 = -100 (mod 41607) (the residue is negative, we add 41607)
-100 + 41607 = 41507 (mod 41607) (now the residue is positive)
Therefore, M = 41507 (mod 41607).
As a result, M is 41507.