Peter notices that the teenage students who are members of the math club have ages whose product is 611520. How many members does the club have?
Extensions: Is it true that whatever number replaces 611520 that this problem has a unique solution? Suppose the club included 12 year olds and the product was
163762560, how many members does the club now have
I tried dividing the given number by number that reflect teenagers.
That is, I started with 13
61150 = 13*47040
47040 = 14*3360
3360=15*224
224=14*16
so the factors are:
13*14*14*15*16
So there are 5 factors, thus 5 members in the club
For the second part, start with 12, then use my method
(hint your number divides by 12 evenly three times)
If you have 12 year olds, you can switch around 12, 16, and 18 year olds. There isn't always a unique solution for if 12 is included.
c) 163 762 560= 2^7*3^9*5*13
1-13 yr-old
2-12 yr-old
3-18 yr-old
1-15 yr-old
=7 members
To find the number of members in the math club, we need to determine the prime factorization of 611520. Then, we can determine how many ways we can distribute the factors to represent the ages of the club members.
First, let's find the prime factorization of 611520:
1. Start by dividing the number by the smallest prime number, which is 2. We get:
611520 ÷ 2 = 305760
2. Next, divide the result by 2 again:
305760 ÷ 2 = 152880
3. Continue dividing by 2 until you can no longer divide evenly:
152880 ÷ 2 = 76440
76440 ÷ 2 = 38220
38220 ÷ 2 = 19110
4. Moving on to the next prime number, divide by 3:
19110 ÷ 3 = 6370
5. Continuing with the prime factorization, divide by 5:
6370 ÷ 5 = 1274
6. Finally, divide by 7 to complete the prime factorization:
1274 ÷ 7 = 182
Now, let's write out the prime factorization of 611520:
611520 = 2^6 × 3 × 5 × 7
To find the number of members, we need to consider the possible ways to distribute these factors among the ages.
For example, if we have 2^3 × 3^2 × 5 × 7, it means that there are 2^3 = 8 members aged 2, 3^2 = 9 members aged 3, 1 member aged 5, and 1 member aged 7. Therefore, the club would have a total of 8 × 9 × 1 × 1 = 72 members.
Now, let's address the extensions given:
1. Is it true that whatever number replaces 611520, this problem has a unique solution?
No, it is not true that whatever number replaces 611520 will have a unique solution. Different prime factorizations may result in different distributions of ages, leading to different final counts of members in the club.
2. Suppose the club included 12-year-olds, and the product was 163762560, how many members does the club now have?
To answer this, we need to determine the prime factorization of 163762560. Following the same process as explained earlier, we get:
163762560 = 2^6 × 3^4 × 5 × 7
Now, we can distribute the factors among the ages:
- There are 2^6 = 64 members aged 2.
- There are 3^4 = 81 members aged 3.
- There is 1 member aged 5.
- There is 1 member aged 7.
- We also have 12-year-olds, which results in an additional 1 member.
Therefore, the club would have a total of 64 × 81 × 1 × 1 × 1 = 5,207 members.