x is a number between 200 and 300. The highest common factor of x and 126 is 21. Find the smallest possible value of x.

Steps to Solve

1. Identify the factors of 126 and their HCF with 21

First, we determine the prime factorization of 126: $$ 126 = 2 \times 3^2 \times 7 $$ Next, we see that the highest common factor (HCF) with 21 should be: $$ 21 = 3 \times 7 $$

2. Determine multiples of 21 in the range 200 to 300

To find the smallest number ( x ) that is a multiple of 21 and lies between 200 and 300, we calculate the smallest multiple of 21 greater than 200: $$ \lceil \frac{200}{21} \rceil = 10 $$ Then we multiply: $$ x = 10 \times 21 = 210 $$

3. Check if the HCF of 210 and 126 is 21

Next, we calculate the HCF of 210 and 126:

The prime factorization of 210 is: $$ 210 = 2 \times 3 \times 5 \times 7 $$

Now, let's find the common factors. The common prime factors are 3 and 7: $$ HCF(210, 126) = 3 \times 7 = 21 $$

Thus, the condition is satisfied.

4. Evaluate the next multiple of 21, if necessary

Check the next multiple of 21: $$ 11 \times 21 = 231 $$ This is greater than 210, so we do not need to check further since we seek the smallest ( x ).