Question

find each n. name the property that you used

1. $3 \times 6 = 6 \times n$
2. $n \times 5 = 0$
3. $365 \times n = 365$
4. $(3 \times 2) \times n = 3 \times (2 \times 10)$
5. $5 \times 6 \times 4 \times 0 = n$
6. $n + 45 = 45$
7. $17 + 62 = 62 + n$
8. $21 + (n + 14) = (21 + 50) + 14$
9. $3 \times (14 - 14) = n$
10. $6 + (9 - n) = 6$
11. reasoning katie says she is thinking of two numbers whose product is 0. john claims that he knows one number for certain. is he right? how does he know?

use the properties of operations to complete.

1. $10 + \underline{\quad\quad} = 10$
2. $(10 \times 6) \times 2 = \underline{\quad\quad} \times (6 \times 2)$
3. $(8 + 2) + 6 = (2 + \underline{\quad\quad}) + 6$
4. $14 + 16 = \underline{\quad\quad} + 14$
5. $c \times d = d \times \underline{\quad\quad}$
6. $(q \times r) \times s = q \times (\underline{\quad\quad} \times s)$

Explanation:

Step1: Solve 3×6=6×n, use commutativity

$3 \times 6 = 6 \times n \implies 18 = 6n \implies n=3$
Property: Commutative Property of Multiplication

Step2: Solve n×5=0, use zero product rule

$n \times 5 = 0 \implies n=0$
Property: Zero Property of Multiplication

Step3: Solve 365×n=365, use identity property

$365 \times n = 365 \implies n=1$
Property: Multiplicative Identity Property

Step4: Solve (3×2)×n=3×(2×10), use associativity

$(3 \times 2) \times n = 3 \times (2 \times 10) \implies 6n = 60 \implies n=10$
Property: Associative Property of Multiplication

Step5: Solve 5×6×4×0=n, use zero property

$5 \times 6 \times 4 \times 0 = n \implies 0 = n$
Property: Zero Property of Multiplication

Step6: Solve n+45=45, use additive identity

$n + 45 = 45 \implies n=0$
Property: Additive Identity Property

Step7: Solve 17+62=62+n, use commutativity

$17 + 62 = 62 + n \implies 79 = 62 + n \implies n=17$
Property: Commutative Property of Addition

Step8: Solve 21+(n+14)=(21+50)+14, use associativity

$21 + (n + 14) = (21 + 50) + 14 \implies n + 35 = 85 \implies n=50$
Property: Associative Property of Addition

Step9: Solve 3×(14-14)=n, simplify first

$3 \times (14 - 14) = n \implies 3 \times 0 = n \implies n=0$
Property: Zero Property of Multiplication

Step10: Solve 6+(9-n)=6, isolate variable

$6 + (9 - n) = 6 \implies 9 - n = 0 \implies n=9$
Property: Additive Inverse Property

Step11: Answer reasoning question

By the Zero Property of Multiplication, if $a \times b = 0$, then at least one of $a$ or $b$ must be 0. So John is right—one of the two numbers must be 0.

Step12: Complete 10+__=10, use additive identity

$10 + 0 = 10$
Property: Additive Identity Property

Step13: Complete (10×6)×2=__×(6×2), use associativity

$(10 \times 6) \times 2 = 10 \times (6 \times 2)$
Property: Associative Property of Multiplication

Step14: Complete (8+2)+6=(2+__)+6, use commutativity

$(8 + 2) + 6 = (2 + 8) + 6$
Property: Commutative Property of Addition

Step15: Complete 14+16=__+14, use commutativity

$14 + 16 = 16 + 14$
Property: Commutative Property of Addition

Step16: Complete c×d=d×__, use commutativity

$c \times d = d \times c$
Property: Commutative Property of Multiplication

Step17: Complete (q×r)×s=q×(__×s), use associativity

$(q \times r) \times s = q \times (r \times s)$
Property: Associative Property of Multiplication

Answer:

1. $n=3$, Commutative Property of Multiplication
2. $n=0$, Zero Property of Multiplication
3. $n=1$, Multiplicative Identity Property
4. $n=10$, Associative Property of Multiplication
5. $n=0$, Zero Property of Multiplication
6. $n=0$, Additive Identity Property
7. $n=17$, Commutative Property of Addition
8. $n=50$, Associative Property of Addition
9. $n=0$, Zero Property of Multiplication
10. $n=9$, Additive Inverse Property
11. John is right. By the Zero Property of Multiplication, if the product of two numbers is 0, at least one of the numbers must be 0, so he knows one number is 0 for certain.
12. $0$ (Additive Identity Property)
13. $10$ (Associative Property of Multiplication)
14. $8$ (Commutative Property of Addition)
15. $16$ (Commutative Property of Addition)
16. $c$ (Commutative Property of Multiplication)
17. $r$ (Associative Property of Multiplication)