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Answer:

Because it would break the Fundamental Theorem of Arithmetic, along with several other important theorems.

Explanation:

Historically, #1# was considered a prime number, but it kept on causing problems, and it was very common for theorems to hold true for "all prime numbers except one".

One very important theorem that wouldn't work if one was considered a prime number is the Fundamental Theorem of Arithmetic. It states that all integers greater than #1# can be represented as a unique product of primes.

The key word there is unique. Let's take #12# as an example integer. Its prime factorization is:
#12=3*2^2#

Suppose you were to allow #1# as a prime number, then you could add #*1# as many times as you wanted to, and it'd still be a product of primes:
#12*3*2^2*1*1...#

This would break the fundamental theorem of arithmetic's property that the prime factorizations are unique. After all, the products would be hardly unique if there were infinitely many.

These types of problems is what eventually led to the exclusion of one as a prime number, because it really only caused problems to the set of prime numbers.

Answer:

See a solution process below:

Explanation:

To round a number, you take the number to the right of where you want the number to be accurate to. In this problem the second decimal place is #color(red)(3)#:

#7.8color(red)(3)65#

The number to the right of it is #color(blue)(6)#:

#7.8color(red)(3)color(blue)(6)5#

If the number to the right is greater than or equal to #5# you add 1 to the number and eliminate all of the other numbers to the right.

If the number to the right is less than #5# you leave the number as is and eliminate all of the other numbers to the right.

In this case #color(blue)(6)# is greater than or equal to #5# so we need to add #1# to #color(red)(3)# and eliminate all the other numbers to the right:

#7.8color(red)(3)color(blue)(6)5# accurate to the second decimal place is #7.8color(red)(4)#

Answer:

4210

Explanation:

A thousand =1000
Since there are 4 thousands, we multiply it by 4 to get #4000#
#4*1000 = 4000#

tens = 10
Since there are 21 tens, we multiply to get the total amount so:
#21*10=210#

Now we add it together
#4000+210=4210#

Answer:

#2#

Explanation:

#16x# is a multiple of #1,2,4,8,16,x,2x,4x,8x,16x#.

#18# is a multiple of #1,2,3,6,9,18#

From here, the common factors of #16x# and #18# are #1# and #2#.

We know that #2>1#,

#:. 2# is the GCF of #16x# and #18#.

Answer:

See a solution process below:

Explanation:

first, using the number 10, because it is 2 digits, we can only use the 1 of the other digits either before or after the 10.

So this would give us: #3 xx 2 = 6#, 3 digit numbers.

710; 810; 910; 107; 108; 109

Then we can combine the 3 single digit numbers in:

#3 xx 2 xx 1 = 6#, additional 3 digit numbers:

#789; 798; 879; 897; 978; 987

Therefore there are a total of 12 three digit numbers you can create.

Answer:

50 seconds

Explanation:

First, let us convert all the information we have to cents and seconds. This will let the calculation be easier.

Every hour has

60 minutes.

So, 5 hours has

#60*5=300# minutes.

And every minute has 60 seconds. So,

300 minutes has

#300*60 = 18000# seconds

In 18 000 seconds, Clare earns $90 (which is 9 000 cents).

If we write it in ratios, we get

Money: time = Money: time

#9 000 : 18 000 = 25 : x#

Now we just need to find #x#.

Remember that,

Product of means = Product of extremes.

So,

#18 000 * 25 = 9000*x#

#450000 = 9000x#

#(450 000)/9000 = x#

#50 = x#

Hence, it would take Clare 50 seconds to earn 25 cents.

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