Lesson Plans SET B for Grades 9-10-Lesson 3 Number Theory HW

Lesson 3 Number Theory HW

Pdf Summary

This set of problems aims to enhance understanding of number theory concepts taught in the Math Kangaroo class. Here's a summary of the solutions provided for each problem:<br /><br />1. The problem asks to find two positive numbers whose square difference is a prime, and the sum of the squares is 1105. The solution identifies the numbers as 23 and 24. This is done by expressing the difference of squares, \((a^2 - b^2 = 47)\), confirming that 47 is a prime, and finding the values that satisfy the sum of \(a^2 + b^2 = 1105\).<br /><br />2. To determine if \(1494 - 714\) is a multiple of 165, the solution involves calculating the expression \((149 - 71)(149 + 71)(149^2 - 71^2)\), then checking divisibility by the factors of 165, which are 3, 5, and 11. Since both 78 and 220 are divisible by these, the expression is confirmed as a multiple of 165.<br /><br />3. The smallest number \(N\) required such that \(15N\) is a perfect square and \(10N\) is a perfect cube is determined to have 72 divisors using prime factorization methods and calculating combinations of exponent increments.<br /><br />4. Given that \(N\) is a 5-digit number with a digit product of 60 and is a multiple of 11, the solution finds possible values of \(N\) as 21516 and 51216, based on constraints given, including checking the reduction by 2 for the product of digits falling to 80.<br /><br />5. Finding a prime number such that \(913 - 1115 = k^2\) involves isolating variables that lead \(p\) to be a factor in different rearrangements. The solution provides a complex manipulation, ultimately leading to the discovery that the prime number \((a = 3)\) makes the expression a perfect square.<br /><br />This concise breakdown encapsulates the methodology applied, ensuring comprehension in preparation for similar exercises.

Keywords

number theory

Math Kangaroo

square difference

prime numbers

divisibility

perfect square

perfect cube

digit product

multiple of 11

prime factorization