Here are the solutions for each problem using prime factorization to find the Least Common Multiple (LCM) and Greatest Common Factor (GCF).
1) 4, 5, 24
*
Prime Factors:
* $4 = 2^2$
* $5 = 5$
* $24 = 2^3 \times 3$
*
LCM: Take the highest power of every prime present: $2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120$.
*
GCF: There are no common factors shared by all three numbers. The GCF is 1.
2) 4, 15, 16
*
Prime Factors:
* $4 = 2^2$
* $15 = 3 \times 5$
* $16 = 2^4$
*
LCM: Highest powers: $2^4 \times 3 \times 5 = 16 \times 15 = 240$.
*
GCF: No common factors shared by all three. The GCF is 1.
3) 32, 34, 36
*
Prime Factors:
* $32 = 2^5$
* $34 = 2 \times 17$
* $36 = 2^2 \times 3^2$
*
LCM: Highest powers: $2^5 \times 3^2 \times 17 = 32 \times 9 \times 17 = 4896$.
*
GCF: The only common factor is 2 ($2^1$). The GCF is 2.
4) 34, 35, 32
*
Prime Factors:
* $34 = 2 \times 17$
* $35 = 5 \times 7$
* $32 = 2^5$
*
LCM: Highest powers: $2^5 \times 5 \times 7 \times 17 = 32 \times 595 = 19040$.
*
GCF: No common factors shared by all three. The GCF is 1.
5) 21, 24, 36
*
Prime Factors:
* $21 = 3 \times 7$
* $24 = 2^3 \times 3$
* $36 = 2^2 \times 3^2$
*
LCM: Highest powers: $2^3 \times 3^2 \times 7 = 8 \times 9 \times 7 = 504$.
*
GCF: The common factor is 3 ($3^1$). The GCF is 3.
6) 10, 16, 21
*
Prime Factors:
* $10 = 2 \times 5$
* $16 = 2^4$
* $21 = 3 \times 7$
*
LCM: Highest powers: $2^4 \times 3 \times 5 \times 7 = 16 \times 105 = 1680$.
*
GCF: No common factors shared by all three. The GCF is 1.
7) 21, 36, 32
*
Prime Factors:
* $21 = 3 \times 7$
* $36 = 2^2 \times 3^2$
* $32 = 2^5$
*
LCM: Highest powers: $2^5 \times 3^2 \times 7 = 32 \times 9 \times 7 = 2016$.
*
GCF: No common factors shared by all three. The GCF is 1.
8) 34, 21, 14
*
Prime Factors:
* $34 = 2 \times 17$
* $21 = 3 \times 7$
* $14 = 2 \times 7$
*
LCM: Highest powers: $2 \times 3 \times 7 \times 17 = 714$.
*
GCF: No common factors shared by all three. The GCF is 1.
9) 35, 15, 12
*
Prime Factors:
* $35 = 5 \times 7$
* $15 = 3 \times 5$
* $12 = 2^2 \times 3$
*
LCM: Highest powers: $2^2 \times 3 \times 5 \times 7 = 4 \times 105 = 420$.
*
GCF: No common factors shared by all three. The GCF is 1.
10) 34, 15, 4
*
Prime Factors:
* $34 = 2 \times 17$
* $15 = 3 \times 5$
* $4 = 2^2$
*
LCM: Highest powers: $2^2 \times 3 \times 5 \times 17 = 4 \times 255 = 1020$.
*
GCF: No common factors shared by all three. The GCF is 1.
Final Answer:
1) LCM: 120, GCF: 1
2) LCM: 240, GCF: 1
3) LCM: 4896, GCF: 2
4) LCM: 19040, GCF: 1
5) LCM: 504, GCF: 3
6) LCM: 1680, GCF: 1
7) LCM: 2016, GCF: 1
8) LCM: 714, GCF: 1
9) LCM: 420, GCF: 1
10) LCM: 1020, GCF: 1
Parent Tip: Review the logic above to help your child master the concept of least common multiple worksheet 6th grade.
