1) Five brothers, each born in a different year, share a gift of $100 according to the following arrangement: each boy, except the youngest, gets $5 more than his next younger brother. How much does the youngest boy get?
2) When I open my math book, two pages face me and the sum of the two page numbers is 317. What is the number of the very next page?
3) 6, 14, and 15 are factors of the natural numbers N. What is the smallest value that N can have?
1) Method 1: The average amount received by the 5 boys is $20. In order of age, the third boy receives the average amount of $20. The youngest receives $10 less than the third boy. Therefore, the youngest receives $10.
Method 2: Let Y represent the amount received by the youngest boy. Then the amounts received are Y, Y + 5, Y + 10, Y + 15, and Y + 20. These amounts can be regrouped and totaled as Y + Y + Y +Y + Y + 5 + 10 + 15 + 20, or more simply as 5Y + 50. Then, 5Y + 50 = 100, 5Y = 50, and Y = 10. The youngest receives $10.
2) The average of the two consecutive page numbers is 317/2 = 158.5. Then the page numbers are 158 and 159, with 159 being the number of the right-hand page. The next page number is 160.
3) Method 1: The LCM of a set of numbers is the smallest number N which each number of the set will divide exactly. LMC(6,14,15) = 210.
Method 2: List the prime factors of each of the given factors of N: 6 = 2 x 3; 14 = 2 x 7, 15 = 3 x 5. N must have 2, 3, 5, and 7 as factors. Then N = 2 x 3 x 5 x 7 = 210.