Whole numbers: primes, GCF, and LCM

Advanced Math 6 · Unit 1 · gcf-lcm-grade6 · Teacher edition

gcf-lcm-grade6

Name ______________________________Date ______________

1.
Write $84$ as a product of primes. How many distinct prime factors does it have?
Answer: ______________
2.
Using $84 = 2^2 \cdot 3 \cdot 7$ and $120 = 2^3 \cdot 3 \cdot 5$ , find $\text{GCF}(84, 120)$ .
Answer: ______________
3.
Using $84 = 2^2 \cdot 3 \cdot 7$ and $120 = 2^3 \cdot 3 \cdot 5$ , find $\text{LCM}(84, 120)$ .
Answer: ______________
4.
Two buses leave the depot together at 8:00. Bus A leaves every $12$ minutes and Bus B every $18$ minutes. When do they next leave together?
1. (A)
  8:06
2. (B)
  8:30
3. (C)
  8:36
4. (D)
  11:36
5.
A ribbon $84$ cm long and a ribbon $120$ cm long are each cut into pieces of the same length, as long as possible, with nothing left over. How many pieces are there in total?
1. (A)
  12
2. (B)
  17
3. (C)
  24
4. (D)
  204
5. (E)
  840

Answer key — Whole numbers: primes, GCF, and LCM

1.
3gcf-lcm-grade6-01
$84 = 2 \times 42 = 2 \times 2 \times 21 = 2^2 \cdot 3 \cdot 7$ . The distinct primes are $2$ , $3$ , and $7$ — 3 of them.
2.
12gcf-lcm-grade6-02
Shared primes: $2$ (lowest power $2^2$ ) and $3$ (lowest power $3$ ). $\text{GCF} = 2^2 \cdot 3 = \mathbf{12}$ .
3.
840gcf-lcm-grade6-03
$\text{LCM} = 2^3 \cdot 3 \cdot 5 \cdot 7 = 8 \cdot 105 = \mathbf{840}$ . Check: $\text{GCF} \times \text{LCM} = 12 \times 840 = 10{,}080 = 84 \times 120$ . ✓
4.
(C)
8:36
gcf-lcm-grade6-04
$\text{LCM}(12, 18) = 2^2 \cdot 3^2 = 36$ . Thirty-six minutes after 8:00 is 8:36. (8:06 uses the GCF, $6$ ; 11:36 uses $12 \times 18 = 216$ minutes, which is a common multiple but not the least.)
5.
(B)
17
gcf-lcm-grade6-05
Piece length $= \text{GCF}(84,120) = 12$ cm. The ribbons give $84 \div 12 = 7$ and $120 \div 12 = 10$ pieces, so $7 + 10 = \mathbf{17}$ pieces. (12 is the piece length, not the count; 204 is the total length.)