22 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!
2 Multiples of 9 ★★★
Write the first six multiples of \(9\).
3 Prime or composite? ★★★
Say whether each number is prime, composite, or neither: \(17\), \(21\), \(29\), \(39\), \(1\), \(51\). Justify composite numbers with a factor.
4 Divisible by 2, 5 or 10 ★★★
For each number, say whether it is divisible by \(2\), by \(5\), by \(10\): \(340\), \(785\), \(1{,}206\), \(4{,}115\).
5 True or false? ★★★
Decide whether each statement is true or false, and explain.
- \(7\) is a factor of \(45\).
- Every multiple of \(6\) is even.
- \(1\) is a prime number.
- A prime number is always odd.
6 Multiples in a range ★★★
List the multiples of \(7\) that are greater than \(40\) and less than \(70\).
7 GCF by listing ★★★
List the factors of \(12\) and of \(30\), then find their GCF.
8 Digit sums ★★★
Use the rules for \(3\) and \(9\) on \(2{,}718\), \(5{,}436\), \(7{,}305\) and \(6{,}481\). Say which are divisible by \(3\) and which by \(9\).
9 Factor tree of 84 ★★★
Draw a factor tree for \(84\) and write its prime factorization with exponents.
10 GCF with prime factors ★★★
Find \(\text{GCF}(56, 42)\) using prime factorizations.
11 LCM of 6 and 15 ★★★
Find \(\text{LCM}(6, 15)\) by listing multiples.
12 Factoring out the GCF ★★★
Rewrite each sum as the GCF times a sum, then evaluate.
- \(45 + 60\)
- \(24 + 40\)
13 Tiling a floor ★★★
A rectangular floor measures \(30\) inches by \(18\) inches (see the figure). A builder wants to cover it with identical square tiles, whole tiles only. What is the largest possible side length of a tile, and how many tiles are needed?
14 Two lighthouses ★★★
Lighthouse A flashes every \(9\) seconds and lighthouse B every \(12\) seconds. They flash together at midnight exactly. After how many seconds do they flash together again?
15 A mystery number ★★★
A number between \(40\) and \(80\) is a multiple of \(4\) and a multiple of \(7\). What is it?
16 Find the mistake ★★★
Sam says: “\(\text{LCM}(8, 12) = 96\) because \(8 \times 12 = 96\).” Explain his mistake and give the correct LCM.
17 School kits ★★★
A teacher has \(72\) pencils and \(54\) erasers. She makes identical kits with no supplies left over. What is the greatest number of kits? How many pencils and erasers are in each kit?
18 Find the other number ★★★
Two numbers have \(\text{GCF} = 4\) and \(\text{LCM} = 60\). One of them is \(12\). Find the other and check your answer.
19 Three school bells ★★★
Three timers ring every \(4\), \(6\) and \(10\) minutes. They all ring together at noon. At what time do they all ring together again?
20 The missing digit ★★★
The number \(5\square4\) has a missing middle digit. Find every digit that makes it divisible by \(9\). Which of those numbers is also divisible by \(4\)?
21 Sums of multiples ★★★
Show that the sum of two multiples of \(6\) is always a multiple of \(6\). Then illustrate with \(42 + 78\) and find \(\text{GCF}(42, 78)\).
22 Hot dogs and buns ★★★
Hot dogs are sold in packs of \(8\) and buns in packs of \(6\). What is the smallest number of hot dogs and buns that gives exactly one bun per hot dog? How many packs of each must you buy?
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