3.1.0: Exercises

Contemporary Mathematics · LibreTexts

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    For the following exercises, use divisibility rules to determine if each of the following is divisible by 2.

    Exercise \(\PageIndex{1}\)

    24

    Exercise \(\PageIndex{2}\)

    37

    Exercise \(\PageIndex{3}\)

    1,345,321

    For the following exercises, use divisibility rules to determine if each of the following is divisible by 3.

    Exercise \(\PageIndex{4}\)

    48

    Exercise \(\PageIndex{5}\)

    210

    Exercise \(\PageIndex{6}\)

    5,345,324

    For the following exercises, use divisibility rules to determine if each of the following is divisible by 5 .

    Exercise \(\PageIndex{7}\)

    130

    Exercise \(\PageIndex{8}\)

    237

    Exercise \(\PageIndex{9}\)

    1,345,321

    For the following exercises, use divisibility rules to determine if each of the following is divisible by 9 .

    Exercise \(\PageIndex{10}\)

    48

    Exercise \(\PageIndex{11}\)

    210

    Exercise \(\PageIndex{12}\)

    5,345,325

    For the following exercises, use divisibility rules to determine if each of the following is divisible by 12.

    Exercise \(\PageIndex{13}\)

    48

    Exercise \(\PageIndex{14}\)

    210

    Exercise \(\PageIndex{15}\)

    5,355,324

    Exercise \(\PageIndex{16}\)

    Determine which of the following numbers are prime: \(\{3,27,77,131,457\}\)

    Exercise \(\PageIndex{17}\)

    Determine which of the following numbers are prime: \(\{31,97,188,389\}\)

    For the following exercises, find the prime factorization of the given number.

    Exercise \(\PageIndex{18}\)

    12

    Exercise \(\PageIndex{19}\)

    53

    Exercise \(\PageIndex{20}\)

    72

    Exercise \(\PageIndex{21}\)

    345

    Exercise \(\PageIndex{22}\)

    938

    Exercise \(\PageIndex{23}\)

    36,068

    Exercise \(\PageIndex{24}\)

    8,211,679

    For the following exercises, find the greatest common divisor of the given set of numbers.

    Exercise \(\PageIndex{25}\)

    \(\{45,245\}\)

    Exercise \(\PageIndex{26}\)

    \(\{11,24\}\)

    Exercise \(\PageIndex{27}\)

    \(\{56,44\}\)

    Exercise \(\PageIndex{28}\)

    \(\{150,600\}\)

    Exercise \(\PageIndex{29}\)

    \(\{1,746,28,324\}\)

    Exercise \(\PageIndex{30}\)

    \(\{30,40,75\}\)

    Exercise \(\PageIndex{31}\)

    \(\{19,45,70\}\)

    Exercise \(\PageIndex{32}\)

    \(\{293,7,298,19,229\}\)

    Exercise \(\PageIndex{33}\)

    \(\{3,927,473,82,709,1,210,121\}\)

    Exercise \(\PageIndex{34}\)

    Make a list of the common divisors of 12 and 18 . What is the GCD of 12 and 18 ? Which of the other common divisors of 12 and 18 divide the GCD?

    Exercise \(\PageIndex{35}\)

    Make a list of the common divisors of 20 and 84 . What is the GCD? Which of the other common divisors of 20 and 84 also divide the GCD?

    Exercise \(\PageIndex{36}\)

    Make a list of the common divisors of 120 and 88 . What is the GCD? Which of the other common divisors of 120 and 88 also divide the GCD?

    Exercise \(\PageIndex{37}\)

    Based on the answers to 34,35 , and 36 , make a conjecture about the GCD of two numbers, and the other common divisors of those numbers.

    Exercise \(\PageIndex{38}\)

    Rebecca wants to cut two lengths of board into equal length pieces, with no leftover piece. The two boards are 230 cm long and 370 cm long. What is the longest length that Rebecca can cut from these boards so that all the cut boards are the same length?

    Exercise \(\PageIndex{39}\)

    Yasmin is playing with her younger brother, Cameron. They are grouping Skittles by color. They have 14 green, 10 yellow, and 8 purple Skittles. Each group must have the same number of green, the same number of yellow, and the same number of purple Skittles. What's the maximum number of piles that Sophia can build with Cameron?

    Exercise \(\PageIndex{40}\)

    Gathii is creating a tile backsplash for his kitchen. He wants to use square tiles to cover a \(330 \mathrm{~cm} \times 12 \mathrm{~cm}\) area. What is the largest size square tile he can use to create this backsplash?

    Exercise \(\PageIndex{41}\)

    Deiji is designing a contest where teams will be given the same number of toothpicks, 5 oz. paper cups, and 2 cm length pieces of string. She has 420 pieces of string, 300 paper cups, and 1,610 toothpicks. What is the maximum number of teams she can have so that every team gets an equal number of pieces of string, paper cups, and toothpicks?

    For the following exercises, find the least common multiple of the given set of numbers.

    Exercise \(\PageIndex{42}\)

    \(\{30,40\}\)

    Exercise \(\PageIndex{43}\)

    \(\{11,24\}\)

    Exercise \(\PageIndex{44}\)

    \(\{14,45\}\)

    Exercise \(\PageIndex{45}\)

    \(\{200,450\}\)

    Exercise \(\PageIndex{46}\)

    \(\{38,077,9,088,687\}\)

    Exercise \(\PageIndex{47}\)

    \(\{36,42,70\}\)

    Exercise \(\PageIndex{48}\)

    \(\{7,13,36\}\)

    Exercise \(\PageIndex{49}\)

    \(\{4,450,864,339,889,157,339\}\)

    Exercise \(\PageIndex{50}\)

    Benjamin and Mia both work at the Grease Fire diner, a local eatery. Benjamin has every 4th day off, and Mia has every 6th day off. How many days pass until they have another day off together?

    Exercise \(\PageIndex{51}\)

    A lunar month is 30 days (rounding off). A new lunar month begins on a Saturday. How many days is it until a lunar month begins on a Saturday again?

    Exercise \(\PageIndex{52}\)

    Isabella is creating a collage for a project and wants a horizontal cut in the collage. The cut will be made by using purple strips of cloth that are 28 mm long, and yellow strips of paper that are 36 mm long. What is the minimum length of the cut can she make using strips with those lengths?

    Exercise \(\PageIndex{53}\)

    Asteroids are objects that orbit the sun. The smallest distance that an asteroid gets to the sun during its orbit is called the perihelion. Asteroids also have orbital periods, or the time it takes to go around the sun exactly one time. The asteroid Ceres has an orbital period (number of days to circle the sun) of 1,680 days. The asteroid Hygiea has an orbital period of 2,031 days. Suppose they are at their perihelion on the same day. How many days will it be before Ceres and Hygiea are at their perihelion on the same day again?

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