To find the least positive number divisible by 700 and 364, we need to find their prime factorizations and then identify the common prime factors.
Step 1: Prime Factorization of 700
Start by dividing 700 by the smallest prime number, which is 2:
700 ÷ 2 = 350
350 ÷ 2 = 175
175 ÷ 5 = 35
35 ÷ 5 = 7
The prime factorization of 700 is 2^2 × 5^2 × 7.
Step 2: Prime Factorization of 364
Now let's find the prime factorization of 364:
364 ÷ 2 = 182
182 ÷ 2 = 91
91 ÷ 7 = 13
The prime factorization of 364 is 2^2 × 7 × 13.
Step 3: Identify Common Prime Factors
To find the least common multiple (LCM), we need to consider each prime factor raised to the highest power found in either factorization. In this case, the common prime factors are 2 and 7, but we only consider the highest powers:
2^2 × 5^2 × 7 × 13
Step 4: Calculate the LCM
To get the least positive number, we multiply all the prime factors raised to their highest powers:
2^2 × 5^2 × 7 × 13 = 4 × 25 × 7 × 13 = 18200
Therefore, the least positive number divisible by 700 and 364 is 18200.