answer:1. Let's denote ( p ) as a prime number and ( a ) and ( b ) as positive integers such that both ( a ) and ( b ) are less than ( p ). We aim to prove that the product ( ab ) is not divisible by ( p ). 2. Considering ( a ) and ( b ) are less than ( p ), this means ( a < p ) and ( b < p ). 3. Assume for the sake of contradiction that ( ab ) is divisible by ( p ), i.e., ( ab equiv 0 pmod{p} ). This can only be true if either ( a equiv 0 pmod{p} ) or ( b equiv 0 pmod{p} ) (this follows from the properties of prime numbers and their factors). 4. Since ( a ) and ( b ) are both strictly less than ( p ), the only integer values they can take are from ( 1 ) to ( p-1 ). A prime number ( p )'s divisors, apart from 1 and itself, must also lie in the range ( 1 ) to ( p-1 ). 5. If ( ab ) were divisible by ( p ), it would imply ( a ) or ( b ) must be a divisor of ( p ) within this range, but no number other than 1 can be divisors of ( p ) within this range, and since ( p ) is prime, it has no other divisors than itself and 1. 6. Given ( a neq p ) and ( b neq p ) (since ( a < p ) and ( b < p )), neither ( a ) nor ( b ) can be zero modulo ( p ), i.e., ( a notequiv 0 pmod{p} ) and ( b notequiv 0 pmod{p} ). 7. This leads to a contradiction as neither ( a ) nor ( b ) can satisfy the condition necessary for ( p ) to divide ( ab ). 8. Therefore, our initial assumption that ( ab ) is divisible by ( p ) must be false. Hence, it must be that ( ab ) is not divisible by ( p ) if both ( a ) and ( b ) are less than ( p ). # Conclusion: [ boxed{text{The product of two positive integers, each less than a prime number } p, text{ is not divisible by } p.} ]

answer:Adding the given equations, we have: [ frac{z(x + y)}{x + y} + frac{x(y + z)}{y + z} + frac{y(z + x)}{z + x} = 5, ] which simplifies to ( x + y + z = 5 ). Subtracting the equations: [ frac{z(y - x)}{x + y} + frac{x(z - y)}{y + z} + frac{y(x - z)}{z + x} = -1. ] Let [ u = frac{x}{x + y} + frac{y}{y + z} + frac{z}{z + x}, quad v = frac{y}{x + y} + frac{z}{y + z} + frac{x}{z + x} ] so ( u + v = 3 ). Also, [ u - v = frac{x - y}{x + y} + frac{y - z}{y + z} + frac{z - x}{z + x} = -1. ] Solving ( u + v = 3 ) and ( u - v = -1 ), we add and subtract these equations to find: [ 2v = 2 Rightarrow v = 1. ] Thus, the value is ( boxed{1} ).

answer:To prove that the sum of the interior angles of a convex polygon with a center of symmetry is divisible by (360^circ), let's start analyzing what we know about polygons with central symmetry. 1. **Recognize Center of Symmetry and Count of Vertices**: A convex polygon with a center of symmetry must have an even number of vertices. We'll denote the polygon as having (2n) vertices. 2. **Formula for Interior Angles**: The sum of the interior angles of any polygon with (V) vertices is given by the formula: [ (V - 2) times 180^circ ] Since our polygon has (2n) vertices, we substitute (V = 2n) into the formula: 3. **Calculate the Sum of Interior Angles**: [ text{Sum of Interior Angles} = (2n - 2) times 180^circ ] 4. **Simplify the Expression**: Simplify the right-hand side of the equation: [ (2n - 2) times 180^circ = 2(n - 1) times 180^circ = 360^circ times (n - 1) ] 5. **Conclusion**: The sum of the interior angles of a convex polygon with a center of symmetry is (360^circ times (n - 1)). Since it can be expressed as a multiple of (360^circ), it is always divisible by (360^circ). Therefore, the sum of the interior angles of such a polygon is divisible by (360^circ). # Conclusion [ boxed{360^circ times (n-1)} ]

answer:To find the equation of the tangent line to the curve y=x^{3}+ln x at x=1, we follow these steps: 1. **Find the derivative of the given function** to determine the slope of the tangent line. The derivative of y=x^{3}+ln x is calculated as follows: [ y' = frac{d}{dx}(x^{3}) + frac{d}{dx}(ln x) = 3x^{2} + frac{1}{x}. ] 2. **Evaluate the derivative at x=1** to find the slope of the tangent line at this point: [ y'(1) = 3(1)^{2} + frac{1}{1} = 3 + 1 = 4. ] 3. **Determine the point of tangency** by substituting x=1 into the original equation: [ y(1) = (1)^{3} + ln(1) = 1 + 0 = 1. ] Therefore, the point of tangency is (1, 1). 4. **Write the equation of the tangent line** using the point-slope form, where the slope is 4 and the point is (1, 1): [ y - 1 = 4(x - 1). ] 5. **Simplify the equation** to get it into the standard form: [ y - 1 = 4x - 4 implies 4x - y - 3 = 0. ] Therefore, the equation of the tangent line to the curve at x=1 is boxed{4x-y-3=0}.