What is the sum of the highest common factor of 100 and 150 together with the common factor of 200 and 250?

The common prime factor is 2. The lowest power of 2 is $2^3$.

The HCF of 24 and 32 is $\boxed{\mathbf{8}}$.

The common prime factor is 2. The lowest power of 2 is $2^5$.

The HCF of 64 and 32 is $\boxed{\mathbf{32}}$.

Common prime factors are 2 and 3. The lowest power of 2 is $2^2$. The lowest power of 3 is $3^1$.

The HCF of 48 and 36 is $\boxed{\mathbf{12}}$.

The common prime factor is 2. The lowest power of 2 is $2^3$.

The HCF of 72 and 32 is $\boxed{\mathbf{8}}$.

Problem solving

1. Factorise 358.

We check for divisibility by prime numbers up to $\sqrt{179} \approx 13.37$.
Primes to check: 3, 5, 7, 11, 13.

• 179 is not divisible by 3 (sum of digits $1+7+9=17$, not divisible by 3).
• 179 is not divisible by 5 (does not end in 0 or 5).
• $179 \div 7 = 25$ with a remainder of 4.
• $179 \div 11 = 16$ with a remainder of 3.
• $179 \div 13 = 13$ with a remainder of 10. Since 179 is not divisible by any prime number up to 13, it is a prime number.

The prime factorization of 358 is $\boxed{\mathbf{2 \times 179}}$.

2. What is the sum of the highest common factor of 100 and 150 together with the highest common factor of 200 and 250?

Prime factorization of 100: $100 = 2^2 \times 5^2$
Prime factorization of 150: $150 = 2 \times 3 \times 5^2$

Prime factorization of 200: $200 = 2^3 \times 5^2$
Prime factorization of 250: $250 = 2 \times 5^3$

The sum is $\boxed{\mathbf{100}}$.