Okay, so I need to figure out how many positive two-digit integers are factors of both 100 and 150. Let me start by understanding the problem. The question is asking for the common two-digit factors of 100 and 150. That means I need to find all the two-digit numbers that divide both 100 and 150 without leaving a remainder.

First, maybe I should find the greatest common divisor (GCD) of 100 and 150. The GCD will give me the largest number that divides both, and then I can find all the divisors of that GCD and check which ones are two-digit numbers. That makes sense because any common factor of 100 and 150 must be a divisor of their GCD.

So, let's compute GCD(100, 150). To do that, I can use the prime factorization method. Let's break down both numbers into their prime factors.

Starting with 100: 100 divided by 2 is 50, divided by 2 again is 25. Then 25 divided by 5 is 5, divided by 5 again is 1. So the prime factors of 100 are 2² × 5².

Now for 150: 150 divided by 2 is 75. 75 divided by 3 is 25. 25 divided by 5 is 5, divided by 5 again is 1. So the prime factors of 150 are 2¹ × 3¹ × 5².

To find the GCD, we take the minimum exponents of the common prime factors. So for prime 2, the minimum exponent is 1 (since 100 has 2² and 150 has 2¹). For prime 5, the minimum exponent is 2 (both have 5²). The other primes (3) are not common, so they don't contribute. Therefore, GCD(100, 150) = 2¹ × 5² = 2 × 25 = 50.

So the greatest common divisor is 50. Now, the common factors of 100 and 150 are exactly the divisors of 50. Let me list all the divisors of 50.

First, the prime factorization of 50 is 2 × 5². The divisors are all combinations of the exponents of the prime factors. The exponents for 2 can be 0 or 1, and for 5 can be 0, 1, or 2. So the divisors are:

- 2⁰ × 5⁰ = 1
- 2⁰ × 5¹ = 5
- 2⁰ × 5² = 25
- 2¹ × 5⁰ = 2
- 2¹ × 5¹ = 10
- 2¹ × 5² = 50

So the divisors of 50 are 1, 2, 5, 10, 25, 50. Now, the question is asking for two-digit positive integers. Two-digit numbers range from 10 to 99. So from the list of divisors, which ones are two-digit numbers?

Looking at the list: 10, 25, 50. The numbers 1, 2, 5 are single-digit, so they don't count. 10 is the first two-digit number, then 25, and 50. After that, the next divisor would be 50, but 50 is already there. The next one would be 50 × 2 = 100, but that's three digits, which is outside the range.

Wait, but 50's divisors include 50 itself. So the two-digit divisors are 10, 25, and 50. Let me count them: 10, 25, 50. That's three numbers. So the answer should be 3.

But let me double-check to make sure I haven't missed any. Let's verify each number:

10: Divides 100? Yes, 100 ÷ 10 = 10. Divides 150? 150 ÷ 10 = 15. So yes.

25: 100 ÷ 25 = 4, 150 ÷ 25 = 6. Both are integers. So yes.

50: 100 ÷ 50 = 2, 150 ÷ 50 = 3. Both are integers. So yes.

Is there any other two-digit number that divides both? Let's check 15. 100 ÷ 15 is approximately 6.666, which is not an integer. 20: 100 ÷ 20 = 5, but 150 ÷ 20 = 7.5, not an integer. 30: 100 ÷ 30 ≈ 3.333, nope. 40: 100 ÷ 40 = 2.5, 150 ÷ 40 = 3.75. No. 45: 100 ÷ 45 ≈ 2.222, nope. 50: already checked. 55: 100 ÷ 55 ≈ 1.818, nope. 60: 100 ÷ 60 ≈ 1.666, nope. 70: 100 ÷ 70 ≈ 1.428, nope. 75: 100 ÷ 75 ≈ 1.333, nope. 80: 100 ÷ 80 = 1.25, nope. 85: 100 ÷ 85 ≈ 1.176, nope. 90: 100 ÷ 90 ≈ 1.111, nope. 95: 100 ÷ 95 ≈ 1.052, nope. 100: three-digit, so no.

So no other two-digit numbers beyond 10, 25, 50 are common factors. Therefore, the answer is 3.

Wait, but let me confirm once more. Since the GCD is 50, all the common factors are divisors of 50, which we listed as 1, 2, 5, 10, 25, 50. Two-digit numbers are 10, 25, 50. So three numbers. That seems correct.

Alternatively, another way to approach this is to list all the two-digit numbers that divide both 100 and 150. But that might take longer. However, since we already used the GCD method, which is systematic and less error-prone, I think that's solid.

Just to recap:

1. Find GCD(100, 150) = 50.
2. Find all divisors of 50: 1, 2, 5, 10, 25, 50.
3. Identify two-digit divisors: 10, 25, 50.
4. Count them: 3.

Therefore, the number of positive two-digit integers that are factors of both 100 and 150 is 3.
Therefore, the number of positive two-digit integers that are factors of both 100 and 150 is $\boxed{3}$.
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