challenge!?

Question

1. place three plus signs and one minus sign tp the left of the equal sign so that the eq.
3 3 4 4 5 5 6 6 7 7 =143 is true
2. seven consecutive integers have a product 32,432,400. find the seven integers.hint: find the prime factors of the product.
3.john asked pete how much money he had on his pocket. pete replied "if i had 12 pesos more than i have and then double that amount , id be 60 pesos richer than i am.how much money had pete?
4.ana gave bikk and clare as much money as each had . then bill gave ana and clare as much money as each had . lastly, clare gave ana and bill as much money as each had . then each had 24 pesos . how much money did each begin with?hint: work bavkward.
5. arrange 2^75, 3^50, and 5^25 from smallest to largest.
6. in the division 24874 divided by 17 which digit should one change in the first number to increase the answer by 100.

Answer 1

2. 32,432,400 = 100 x 324324 = 10 x 10 x 324 x 1001
= 2 x 2 x 5 x 5 x 3 x 108 x 77 x 13
= 2 x 2 x 5 x 5 x 3 x 12 x 9 x 7 x 11 x 13
= 2 x 2 x 5 x 5 x 3 x 2 x 2 x 3 x 9 x 7 x 11 x 13
= 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 5 x 5 x 7 x 11 x 13

At this point, it's obvious the answer revolves around 11 and 13 (as they are both prime). After some trial and error, I figured it out. The answer is

15 x 14 x 13 x 12 x 11 x 10 x 9

15 = 3 x 5
14 = 2 x 7
13
12 = 2 x 2 x 3
11
10 = 2 x 5
9 = 3 x 3

5) Arrange 2^(75), 3^(50), 5^(25) from smallest to largest.

2^(75) = (2^25)^3
3^(50) = (3^25)^2
5^(25) = (5^25)^1

It's a fact that
5 < 8 < 9.

f(x) = x^(25) is an increasing function (that is, if a < b, then f(a) < f(b). That means

f(5) < f(8) < f(9)
5^25 < 8^25 < 9^25

But 8 = 2^3 and 9 = 3^2, so

5^25 < [2^3]^25 < [3^2]^25

Whenever we have a power to a power, we multiply exponents.

5^25 < 2^(75) < 3^(50)

And that's our order from smallest to largest.

6)

We can solve this using the algebra guidelines below
We know that 24874/17 is some value,

24874/17 = n

And we know that some unknown number divided by 17 should equal that number plus 100.

a/17 = n + 100

All we have to do is solve for a.

a = 17n + 1700
a = 17(24874/17) + 1700
a = 24874 + 1700
a = 26574

I have two digits that had to be changed; I maybe misinterpreted the answer or I could be wrong.

Answer 2

2. Seven consecutive integers have a product 32,432,400. Find the seven integers.hint: Find the prime factors of the product.

32,432,400^(1/7) = 11.83
The nearest integer is 12 so it is probably the middle digit of the seven consecutive integers. So pick:

9*10*11*12*13*14*15 = 32,432,400

The product checks, so it is correct. Those are the seven consecutive numbers.

No need to find the prime factors.

3. John asked pete how much money he had on his pocket. pete replied "if i had 12 pesos more than i have and then double that amount , id be 60 pesos richer than i am.how much money had pete?

Let x be the amount of money Pete had.

2(x + 12) = x + 60
2x + 24 = x + 60
x = 36 pesos

4. Ana gave bikk and clare as much money as each had . then bill gave ana and clare as much money as each had . lastly, clare gave ana and bill as much money as each had . then each had 24 pesos . how much money did each begin with?hint: work bavkward

I assume bikk is supposed to be Bill.

To start we have the three individuals with the following amounts of money.

Ana = x
Bill = y
Clare = z

After round 1 with Ana giving we have

Ana = x - y - z
Bill = 2y
Clare = 2z

After round 2 with Bill giving we have

Ana = 2x - 2y - 2z
Bill = 2y - (x - y - z) - 2z = -x + 3y - z
Clare = 4z

After round 3 with Clare giving we have

Ana = 4x - 4y - 4z = 24
Bill = -2x + 6y - 2z = 24
Clare = 4z - (2x - 2y -2z) - (-x + 3y - z) = -x - y + 7z = 24

Now we have three equations in three unknowns. These equations solve easily to the original amounts being:

Ana = x = 39
Bill = y = 21
Clare = z = 12

No need to work backwards.

5. Arrange 2^75, 3^50, and 5^25 from smallest to largest.

First let's compare 2^75 and 3^50.

3^x = 2^75
ln 3^x = ln 2^75
x ln 3 = 75 ln 2
x = (75 ln 2) / ln 3
x = 47.3 < 50

So 2^75 < 3^50

Next let's compare 2^75 and 5^25.

5^x = 2^75
ln 5^x = ln 2^75
x ln 5 = 75 ln 2
x = (75 ln 2) / ln 5
x = 32.3 > 25

So 2^75 > 5^25

So the final order is:

5^25 < 2^75 < 3^50

Answer 3

6. Consider long division for 24874/17:

Start with the first two digits - 24
24 = 17*1 + 7
Since the remainder is 7, you would then drop down the 8 to get the next two digits - 78
78 = 17*4 + 10
Since the remainder is 10, you would then drop down the 7 to get the next 3 digits - 107
107 = 17*6 + 5
Since the remainder is 5, you would then drop down the 4 to get the next 2 digits - 54
54 = 17*3 + 3
Since there are no more digits to drop, your final remainder is 3.

Thus, 24874/17 = 1463 r.3

To increase this number by 100, you want to consider changing the second and third digits so that your number becomes 26574.

Start with the first two digits - 26
26 = 17*1 + 9
Since the remainder is 9, you would then drop down the 5 to get the next two digits - 95
95 = 17*5 + 10

We are back to a remainder of 10 for the third step, so the rest of the answer is the same as before.

Then 26574/17 = 1563 r.3, which is exactly 100 more than the previous answer.

Answer 4

1. 3 3 + 4 4 + 5 5 - 6 6 + 7 7 = 143
2. 32,432,400.= (15)(14)(13)(12)(11)(10)(9)

I don't have time. Hopefully, someone else can finish the remaining problems.

---------
I got the second one by permutation nP7 = 32,432,400 after I found 13 is a factor.

Answer 5

ummmmm...42!