Example Question - factors of 72
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Calculating Probability with Divisors and Factors
To solve this problem, we need to determine the divisors of 40 and the factors of 72, then calculate the probability of rolling one of the divisors first, and then calculate the probability of rolling one of the factors afterwards. Given that this appears to involve dice (as indicated by the image of a die), we will assume a standard 6-sided die for our calculations. **Step 1: Divisors of 40** The divisors of 40 are the numbers which divide 40 without leaving a remainder. These are: 1, 2, 4, 5, 8, 10, 20, and 40. However, since we are dealing with a standard die, we are only interested in the divisors that are between 1 and 6 (inclusive), because those are the only outcomes possible with a single roll of the die. The divisors of 40 that fall within this range are: 1, 2, 4, and 5. **Probability of rolling a divisor of 40:** There are 4 favorable outcomes (1, 2, 4, 5) out of 6 possible outcomes (1-6 on a die), so the probability is: P(divisor of 40) = 4/6 = 2/3 after simplifying. **Step 2: Factors of 72** The factors of 72 are the numbers which can be multiplied by another number to get 72. These are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. Again, considering we have a standard die, we need the factors that are between 1 and 6. These factors are: 1, 2, 3, 4, and 6. **Probability of rolling a factor of 72:** There are 5 favorable outcomes (1, 2, 3, 4, 6) out of 6 possible outcomes, so the probability is: P(factor of 72) = 5/6. **Step 3: Combined probability** To find the combined probability of both events happening (rolling a divisor of 40 and then a factor of 72), we need to multiply the probabilities of each individual event: P(total) = P(divisor of 40) * P(factor of 72) P(total) = (2/3) * (5/6) Multiplying the fractions, we get: P(total) = (2*5) / (3*6) P(total) = 10 / 18 P(total) = 5 / 9 after simplifying. So, the simplified probability of rolling a divisor of 40 and then rolling a factor of 72 with a standard die is 5/9.
Calculating the Probability of Rolling Specified Numbers on a Die
To solve the problem, we first need to identify the divisors of 40 and the factors of 72, and then calculate the probability of rolling each. The divisors of 40 (numbers that can divide 40 without leaving a remainder) are: 1, 2, 4, 5, 8, 10, 20, and 40. If we assume the die is a standard six-sided die, then the only divisors of 40 that are possible to roll are 1, 2, 4, and 5. Thus, there are 4 favorable outcomes out of 6 possible outcomes when rolling a divisor of 40. Now let's find the factors of 72 (numbers that 72 can be divided by without leaving a remainder), which are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. Since a standard die only has six sides, the only factors of 72 that can appear on a die roll are 1, 2, 3, 4, and 6. This means there are 5 favorable outcomes out of 6 possible outcomes when rolling a factor of 72. Now, to find the combined probability of these two independent events, we multiply the probabilities of each event occurring: Probability of rolling a divisor of 40: 4/6 (because there are 4 possible divisors on a die out of 6 sides) Probability of rolling a factor of 72: 5/6 (because there are 5 possible factors on a die out of 6 sides) Combined probability = (4/6) * (5/6) Let's simplify this fraction: (4 * 5) / (6 * 6) = 20 / 36 Now we simplify the fraction by dividing both the numerator and the denominator by the greatest common divisor, which is 4: 20 / 36 = (20÷4) / (36÷4) = 5 / 9 So the probability of rolling a divisor of 40 and then rolling a factor of 72 is 5/9 when using a standard six-sided die.
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