Question

The largest number that divides 450, 577 and 704, leaving remainders 9, 10 and 11 respectively, is 

(A) 63 

(B) 577 

(C) 450 

(D) More than one of the above 

(E) None of the above

Answer 1

Correct option is (A) 63  

The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides each of the given integers without leaving a remainder.

To find the largest number that divides three numbers, leaving specified remainders, we can use the concept of the Chinese Remainder Theorem.

Let N be the largest number we are looking for.

According to the given information:

1. N divides (450 - 9) = 441 evenly because 450 leaves a remainder of 9 when divided by N.

2. N divides (577 - 10) = 567 evenly because 577 leaves a remainder of 10 when divided by N.

3. N divides (704 - 11) = 693 evenly because 704 leaves a remainder of 11 when divided by N.

Now, we need to find the largest number that divides all three of these numbers (441, 567, and 693) evenly.

To find the greatest common divisor (GCD) of these three numbers:

GCD(441, 567, 693) = GCD(GCD(441, 567), 693)

Now, calculate the GCD of 441 and 567:

GCD(441, 567) = 63

Now, find the GCD of 63 and 693:

GCD(63, 693) = 63

So, the largest number N that divides 450, 577, and 704, leaving remainders 9, 10, and 11 respectively, is 63.

∴ The largest number is 63.