Geometry Puzzle – Critical Thinking 19
If N and P are distinct primes, how many positive divisors does N^2 times P^4 have? The slick way: for any positive integer factored as p1^a * p2^b * … * pk^z, the divisor count is (a+1)(b+1)…(z+1). Plug in: (2+1)(4+1) = 15. Key takeaways: Divisor count formula: if \( n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} […]
Geometry Puzzle – Critical Thinking 16
Two sides of a triangle are 10 and 5. What areas are possible? The triangle inequality forces the third side strictly between 5 and 15. The area is at its largest when the included angle between the two given sides is 90 degrees, giving a maximum area of (1/2)(10)(5) = 25 square units. Key takeaways: […]
