Below
Step-by-step explanation:
6)
The sum that we have is 85 +99
We want to express it as the product of a whole number thar is greater than 1 and a sum of two whole numbers.
Notice that: 85 = 84+1
β 85 + 99 = 84 + 1 + 99 = 84 + 100
84 and 100 are even numbers so we can factor using 2.
β 84 + 100 = 2(42 +50)
2 is greater than one and 42+50 is the sum of two whole numbers so all the conditions are satisfied.
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7)
Dasha go on business trips every 9 months while Charlie go every 6 months.
They came back at the same time.
So Charlie has to wait 6 months before going and Dasha nine months.
Dasha will be alone home for 3 months so she doesn't need to hire someone.
Here is what happens:
β Both Dasha and Charlie are home.
β After 6 months Charlie go and Dasha is at home
β after 3 months Dasha goes also and Charlie is home
β after 3 months charlie go and Dasha is home
β after 3 months both are home.
β aftet 3 months they both go
So the period is:
β 6+3+3+3+3 = 18
So after 18 months they should hire someone.
The picture below makes the understanding easier. ( x is Charlie and y is Dasha)
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8)
Pime factorisation
β 96Γ·2= 48
β 48Γ·2= 24
β 24 Γ· 2 = 12
β 12 Γ· 2 = 6
β 6Γ·2 = 3
β 3Γ·3 = 1
=> 96 = 2 Γ 2 Γ2Γ2Γ2 Γ3 =2^5 Γ3
β 80Γ· 2 = 40
β 40 Γ· 2 = 20
β 20Γ·2 = 10
β 10Γ·2 = 5
β 5Γ·5 = 1
=> 80 = 2Γ2Γ2Γ2Γ5 = 2^4 Γ 5
So the GCF is 2^4 wich is 16
He can make 16 party
β 80Γ·16 = 5
There will be 5 boxes of raisin in each one
β 96Γ·16 = 6
There will be 6 pencils in each party