a two-digit number is a perfect square and has exactly 9 positive integral factors. What is the number?
How would i figure this out?
I would start with two digit numbers...
100 factors 2,5,25,4,50,10, 100, 1 eight factors check that.
81 factors 3,9,27,81,1six factors
64 2,4,16,32, 64,1,8 seven factors
recheck those, and work downward to find one with nine factors.
There aren't that many two-digit perfect squares. I'd simply try them all out, starting with the largest I can think of (since that's most likely to have nine factors) and work backwards until I find one that works.
To find a two-digit number that is a perfect square and has exactly 9 positive integral factors, we can follow these steps:
Step 1: List all two-digit perfect squares: A two-digit perfect square is a number between 10 and 99 whose square root is a whole number. These numbers are: 16, 25, 36, 49, 64, and 81.
Step 2: Find the factors of each two-digit perfect square: To determine the number of factors, we need to list the factors of each perfect square and count them. The factors of each perfect square are:
16: 1, 2, 4, 8, 16 (total factors = 5)
25: 1, 5, 25 (total factors = 3)
36: 1, 2, 3, 4, 6, 9, 12, 18, 36 (total factors = 9)
49: 1, 7, 49 (total factors = 3)
64: 1, 2, 4, 8, 16, 32, 64 (total factors = 7)
81: 1, 3, 9, 27, 81 (total factors = 5)
Step 3: Identify the two-digit perfect square with exactly 9 positive integral factors: From the list, we can see that the only perfect square with exactly 9 factors is 36.
Therefore, the two-digit number that is a perfect square and has exactly 9 positive integral factors is 36.
To figure out the two-digit number that is a perfect square and has exactly 9 positive integral factors, we can follow these steps:
1. Find the perfect squares between 10 and 99 (all two-digit numbers).
- Starting with 10, square each number until you reach 99. For example, the perfect squares between 10 and 99 are 16, 25, 36, 49, 64, 81.
2. Determine the number of factors for each perfect square.
- To find the factors of a perfect square, count the number of pairs of factors. For example, the perfect square of 16 has 5 pairs of factors: (1, 16), (2, 8), and (4, 4). It has a total of 9 factors, but only 5 distinct positive integral factors.
3. Identify the perfect square that has exactly 9 positive integral factors.
- After checking all the perfect squares between 10 and 99, you will find that only one perfect square, which is 36, has exactly 9 positive integral factors.
Therefore, the two-digit number that is a perfect square and has exactly 9 positive integral factors is 36.