and  are positive integers such that , where  is a prime number.
The number of pairs  is ... ?

If positive integers p, q and p + q are all prime, what is the least possible value of pq?

For each positive integer n ≤ 55, if S(n) is the sum of the positive integer divisors of n, what is the greatest possible value of S(n)?

The perimeter of an isosceles triangle is 40 centimeters. The congruent side lengths are integers. What is the maximum number of centimeters in the length of the third side?

Donny needs to find the surface area of a dozen donuts so he can make enough glaze. He doesn’t know how to calculate the exact surface area of a donut, so he makes an approximation based on a roll of 1-inch-wide masking tape, which has the same volume as the donut. The outer radius of the tape is 2.5 inches, and the inner radius is 1.5 inches. What surface area did Donny estimate for a dozen donuts? Express your answer in terms of π.

A positive integer is divided by 5. The quotient and the remainder are recorded.
The same number is divided by 3. Again the quotient and the remainder are
recorded. If the same two numbers in different order are recorded, find the
product of all possible values of the original number

Prove that if \(a\ge3\), \(b\ge3\), \(a^2+b^2\ge25\) then \(a+b\ge7\)

For grade 5 students: A car goes from A to B with a speed at 45km per hour, but at the same time, an another car goes from B to A with a speed at 50km per hour. After 45 minutes, the two cars meet at each other. Find the length of the line segmet AB (Here's the hint: v(speed) = s(length) : t(time)

Length of the line segment AB: ........... km

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A fraction, expressed in its lowest terms \(\dfrac{a}{b}\) , can also be written in the form  \(\dfrac{2}{n}+\dfrac{1}{n^2}\) , where n is a positive integer. If a + b = 1024, what is the value of a?