When a number is divided by 7 and 9 , the remainder is 4 and 3. what will be the remainder of a same no. when it is divided by 37.
To find the remainder when a number is divided by 37, we can use the concept of the Chinese Remainder Theorem. Here's how you can solve it:
Step 1: Let's consider a number that gives a remainder of 4 when divided by 7, and a remainder of 3 when divided by 9.
Step 2: First, we need to find two numbers, m1 and m2, such that m1 is divisible by 7 and 9, and m2 is divisible by 37.
Step 3: By observing the given two remainders (4 and 3), we can see that one possible solution is m1 = 7 * 9 = 63 and m2 = 37.
Step 4: Next, we need to find the inverses of m1 and m2 modulo 7 and 9 respectively.
For m1 = 63:
Using the Extended Euclidean Algorithm, we can find the inverse of 63 modulo 7, which is 1.
For m2 = 37:
Using the Extended Euclidean Algorithm, we can find the inverse of 37 modulo 9, which is 4.
Step 5: Finally, we can calculate the remainder by using the formula:
Remainder = [(Number * m1 * 1) + (Number * m2 * 4)] % (7 * 9 * 37)
Step 6: Since the value of m1 is 63 and the value of m2 is 37, we can simplify the formula:
Remainder = [(Number * 63) + (Number * 37 * 4)] % (7 * 9 * 37)
Remainder = [63Number + 148Number] % 2331
Step 7: Simplifying further:
Remainder = (211Number) % 2331
So the remainder when the number is divided by 37 is given by (211Number) % 2331.
Note: To find the specific remainder when dividing a number by 37, you would need to have a specific value for the number.