Factors and Multiples: math test solutions, Grade 6 – download the PDF

1 Vocabulary and lists / 4 pts

1. \(1 \times 40\), \(2 \times 20\), \(4 \times 10\), \(5 \times 8\): the factors are \(1, 2, 4, 5, 8, 10, 20, 40\). (1 pt)
2. \(15, 30, 45, 60, 75\). (1 pt)
3. Composite, because \(57 = 3 \times 19\) (its digit sum \(12\) is divisible by \(3\)). (1 pt)
4. \(23\), \(43\) and \(53\) are prime; \(33 = 3 \times 11\) is composite. (1 pt)

2 Divisibility / 3 pts

1. It ends in \(6\): divisible by \(2\). The digit sum is \(7 + 2 + 3 + 6 = 18\): divisible by \(3\) and \(9\), hence by \(6\). The last two digits \(36 = 4 \times 9\): divisible by \(4\). It does not end in \(0\) or \(5\): not divisible by \(5\) or \(10\). So \(2, 3, 4, 6, 9\) divide it. (2 pts)
2. For \(5\), the last digit is \(0\) or \(5\). The digit sum \(4 + 8 + \square = 12 + \square\) must be a multiple of \(3\): \(\square = 0\) works (sum \(12\)), \(\square = 5\) does not (sum \(17\)). The number is \(480\). (1 pt)

3 Prime factorization / 3 pts

1. \(180 = 18 \times 10 = (2 \times 3 \times 3) \times (2 \times 5)\), so \(180 = 2^2 \times 3^2 \times 5\). (2 pts)
2. \(12 = 2^2 \times 3\) and both \(2^2\) and \(3\) appear in the factorization of \(180\), so \(180 = 12 \times 15\). (1 pt)

4 GCF and LCM / 4 pts

\(45 = 3^2 \times 5\) and \(75 = 3 \times 5^2\).

1. Smaller exponents: \(3 \times 5 = 15\), so \(\text{GCF} = 15\). (2 pts)
2. Larger exponents: \(3^2 \times 5^2 = 225\), so \(\text{LCM} = 225\). Check: \(15 \times 225 = 3{,}375 = 45 \times 75\). (2 pts)

5 Distributive property / 2 pts

\(\text{GCF}(54, 90) = 18\) (\(54 = 2 \times 3^3\), \(90 = 2 \times 3^2 \times 5\)). (1 pt)

\(54 + 90 = 18(3 + 5) = 18 \times 8 = 144\). (1 pt)

6 Word problems / 4 pts

1. \(84 = 2^2 \times 3 \times 7\) and \(108 = 2^2 \times 3^3\), so \(\text{GCF} = 2^2 \times 3 = 12\). She sets up \(12\) stations with \(84 \div 12 = 7\) beanbags and \(108 \div 12 = 9\) jump ropes each. (2 pts)
2. \(\text{LCM}(15, 20) = 60\) (\(15 = 3 \times 5\), \(20 = 2^2 \times 5\), so \(2^2 \times 3 \times 5 = 60\)). They are together again \(60\) minutes later, at 7:00 a.m. (2 pts)