Answer: the remainder when n is divided by 57 is 3.
Step-by-step explanation: Let's first consider the relation between the divisors.
We can see that both 1995 and 57 are factors of 113,235, which is their LCM. Therefore, any integer that leaves a remainder of 75 when divided by 1995 must also leave the same remainder of 75 when divided by 57.
To see why, suppose that n leaves a remainder of 75 when divided by 1995. Then we can write:
n = 1995q + 75
where q is the quotient of the division. We can rearrange this equation as:
n = (57 x 35)q + 75
n = 57(35q + 1) - 2(1995q + 40)
The first term on the right-hand side is a multiple of 57, so n leaves a remainder of -2(1995q + 40) when divided by 57. However, we can add a multiple of 57 to this remainder without changing the remainder mod 57. Adding 2(113,235) = 226,470 to -2(1995q + 40), we get:
-2(1995q + 40) + 226,470 = -2(1995q - 112,215)
Therefore, n leaves a remainder of -2(1995q - 112,215) when divided by 57. However, since n leaves a remainder of 75 when divided by 1995, we can also write:
n = 1995p + 75
for some integer p. Substituting this into the above equation and simplifying, we get:
75 - 2(1995q - 112,215) = 57r
where r is the remainder we are trying to find. Solving for r, we get:
r = 3
Therefore, the remainder when n is divided by 57 is 3.
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Answer: the remainder when n is divided by 57 is 3.
Step-by-step explanation:
Let's first consider the relation between the divisors.
We can see that both 1995 and 57 are factors of 113,235, which is their LCM. Therefore, any integer that leaves a remainder of 75 when divided by 1995 must also leave the same remainder of 75 when divided by 57.
To see why, suppose that n leaves a remainder of 75 when divided by 1995. Then we can write:
n = 1995q + 75
where q is the quotient of the division. We can rearrange this equation as:
n = (57 x 35)q + 75
n = 57(35q + 1) - 2(1995q + 40)
The first term on the right-hand side is a multiple of 57, so n leaves a remainder of -2(1995q + 40) when divided by 57. However, we can add a multiple of 57 to this remainder without changing the remainder mod 57. Adding 2(113,235) = 226,470 to -2(1995q + 40), we get:
-2(1995q + 40) + 226,470 = -2(1995q - 112,215)
Therefore, n leaves a remainder of -2(1995q - 112,215) when divided by 57. However, since n leaves a remainder of 75 when divided by 1995, we can also write:
n = 1995p + 75
for some integer p. Substituting this into the above equation and simplifying, we get:
75 - 2(1995q - 112,215) = 57r
where r is the remainder we are trying to find. Solving for r, we get:
r = 3
Therefore, the remainder when n is divided by 57 is 3.
Answer:
We can use the Chinese Remainder Theorem to solve this problem.
First, we note that since the remainder when n is divided by 1995 is 75, we can write:
n = 1995q + 75, where q is an integer.
Next, we want to find the remainder when n is divided by 57. To do this, we can use the fact that:
1995 ≡ 27 (mod 57)
75 ≡ 18 (mod 57)
Substituting these congruences into our expression for n, we get:
n ≡ 1995q + 75 ≡ 27q + 18 (mod 57)
So the remainder when n is divided by 57 is the same as the remainder when 27q + 18 is divided by 57.
To find this remainder, we can test different values of q. For example, when q = 1, we have:
27q + 18 = 27(1) + 18 = 45
Since 45 is not divisible by 57, the remainder when n is divided by 57 is 45.
Therefore, if an integer n is divided by 1995 and the remainder is 75, then the remainder when n is divided by 57 is 45.