Find the largest number which divides 545 and 725 without leaving any remainder.
The correct answer and explanation is:
To find the largest number that divides both 545 and 725 without leaving a remainder, you need to find the Greatest Common Divisor (GCD) of these two numbers.
Steps to find the GCD of 545 and 725:
- Prime Factorization of 545:
- Start by dividing 545 by the smallest prime numbers.
- 545 is an odd number, so it’s not divisible by 2.
- Divide 545 by 5 (since it ends in 5), so: 545÷5=109545 \div 5 = 109
- Now, check if 109 is divisible by any prime numbers. It is not divisible by 2, 3, or 5, and since the square root of 109 is approximately 10.44, we only need to check prime numbers up to 11. 109 is a prime number, so the prime factorization of 545 is: 545=5×109545 = 5 \times 109
- Prime Factorization of 725:
- Divide 725 by 5 (since it ends in 5): 725÷5=145725 \div 5 = 145
- Next, divide 145 by 5 again: 145÷5=29145 \div 5 = 29
- 29 is a prime number, so the prime factorization of 725 is: 725=52×29725 = 5^2 \times 29
- Finding the GCD:
- The prime factorization of 545 is 5×1095 \times 109.
- The prime factorization of 725 is 52×295^2 \times 29.
- The only common prime factor between 545 and 725 is 5. The lowest power of 5 in both factorizations is 515^1.
Thus, the GCD of 545 and 725 is 5.
Explanation:
The GCD of two numbers is the largest number that divides both of them exactly. In this case, both 545 and 725 have 5 as a common factor, and since 5 is the only factor they share, it is the greatest common divisor. Using prime factorization, we broke down both numbers into their prime factors and identified the common factors to find the GCD.