Exercise 42 Page 32

Consider that a radio station gives away 15 dollars to every 15th caller, 25 dollars to every 25th caller, and a free concert ticket to every 100th caller. Let's make three lists that represent the caller who will win each prize throughout the day. Winner of $15 &= 15,30,45,60,75,90,105,120,... Winner of$25 &= 25,50,75,100,125,150,175,... Winner of ticket&= 100,200,300,400,500,600,... To find when the station will first give away all three prizes to one caller, we can find the least common multiple (LCM) of the three caller numbers. To do so, recall that the LCM of two numbers is the smallest whole number that is a multiple of both numbers. To find the LCM, we can follow two steps.

• Express each number as a product of its prime factors.
• Multiply the highest power of each prime factor.

  Let's start by finding the prime factors with the factor tree of each number.

  
  The red circles are the prime factors of the given number. We can use this factors to write the prime factorization of each number. 15 &= 3 * 5 25 &= 5 * 5 = 5^2 100 &= 2 * 2 * 5 * 5 = 2^2 * 5^2 Let's identify the highest power of each prime factor. 15 &= 3 * 5 25 &= 5^2 100 &= 2^2 * 5^2 Finally, we can multiply these powers.
  3* 5^2 * 2^2

  CalcPow

  Calculate power

  3 * 25 * 4

  Multiply

  Multiply

  300
  Therefore, the 300th caller will win the three prizes. Now we need to find the total amount of money and number of tickets that the radio station will have given away when the 300th caller win the prizes. Let's find this information one piece at a time!

  Money

  Since every 15th caller wins $15, we will need to find how many times this has occurred by the 300th caller. To do so, we can divide 300 by 15.
  300

  Div

  Divide by 15

  300 ÷ 15

  CalcQuot

  Calculate quotient

  20
  This means that 20 people receive $ 15. Then, the total amount of money given from this prize is the product of 20 multiplied by $ 15. 20 * $ 15 = $ 300 We can follow a similar process for the other cash prize, this time with every 25th caller winning $ 25. To find the number of times this has occurred by the 300th caller, we can divide 300 by 25.
  300

  Div

  Divide by 25

  300 ÷ 25

  CalcQuot

  Calculate quotient

  12
  We found that 12 people will have received $ 25. Let's multiply 12 by 25 to find the total amount of money given from this prize. 12 * $ 25 = $ 300 To find the total amount of money given by the time the 300th person calls in, we can add the amounts of money given for the $15 and $25 prizes. $300 + $300 =$ 600 The station will have given away $600 when the 300th caller wins.

  Tickets

  Every 100th caller receives a free concert ticket. This means that the number of times this has happened by the time the 300th caller wins is 3. 300/100=3 Therefore, three people have received a concert ticket.