When a number is divided by 42 the remainder is 39 and when it is divided by 36 the remainder is 33.find the number.
To find the number, we can use the concept of modular arithmetic.
Let's say the number we're trying to find is represented by "x". According to the problem, when x is divided by 42, the remainder is 39, and when it is divided by 36, the remainder is 33.
In modular arithmetic, whenever we divide a number by a divisor, the remainder represents the number's position within that modulus. So, we can set up the following modular equations based on the problem:
x ≡ 39 (mod 42) - Equation 1
x ≡ 33 (mod 36) - Equation 2
Now, we need to solve these modular equations simultaneously to find the value of x. We can use the Chinese Remainder Theorem (CRT) to solve such equations.
First, let's identify a number "m" which is the product of the two divisors, namely 42 and 36. So, m = 42 * 36 = 1512.
Next, we need to find two numbers, "a" and "b", such that their product with the respective moduli leads to 1. In other words, we want a * 42 ≡ 1 (mod 36) and b * 36 ≡ 1 (mod 42).
Using the Extended Euclidean algorithm, we can find that a = 23 and b = 7 satisfy these congruences.
Now, we can use the CRT to find the value of x:
x ≡ (39 * 23 * 36 + 33 * 7 * 42) (mod 1512)
x ≡ 332031 (mod 1512)
To find the actual value of x, we need to take this result and divide it by 1512, and then consider the remainder.
x = 332031 % 1512
x = 159
Therefore, the number we're looking for is 159.