Questions on Algebra: Divisibility and Prime Numbers answered by real tutors!

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Question 992876: The area of a rectangle is 991 cm2. If 991 is a prime number, what are the whole number dimensions of the rectangle?
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The area of a rectangle is 991 cm2. If 991 is a prime number, what are the whole number dimensions of the rectangle?
-----------------
991 is a prime number, there's no "if" about it.
What 2 integers have a product of 991?
---
What is the definition of prime?

Question 992873: The consecutive numbers 2 and 3 are prime. How do you know that there are no other consecutive prime numbers?
Found 2 solutions by stanbon, Alan3354:
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The consecutive numbers 2 and 3 are prime. How do you know that there are no other consecutive prime numbers?
---
Ans:: If they are consecutive one must be even and the other odd.
But the only even prime is "2". So there cannot be another
consecutive pair of primes.
Cheers,
Stan H.
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Question 992079: Find the smallest number that is divisible by first 10 natural numbers.
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2 X 2 X 2 X 3 X 3 X 5 X 7

2 makes it divisible by 2
3 makes it divisible by 3
2 X 2 makes it divisible by 4
5 makes it divisible by 5
2 X 3 makes it divisible by 6
7 makes it divisible by 7
2 X 2 X 2 makes it divisible by 8
3 X 3 makes it divisible by 9
2 X 5 makes it divisible by 10

John

My calculator said it, I believe it, that settles it

Question 990234: What is the smallest whole number divisible by every whole number from 1 to 10.
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.
8*9*5*7 = 2520.

Question 990012: Nina baked 96 cookies while Jean baked 120. They want to put the biggest number of cookies in each plastic bag so that both have the same number of cookies in a bag wherein there are no left over cookies and their cookies are separate. Can you help them find how to put cookies into plastic bags?
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Nina baked 96 cookies while Jean baked 120. They want to put the biggest number of cookies in each plastic bag so that both have the same number of cookies in a bag wherein there are no left over cookies and their cookies are separate. Can you help them find how to put cookies into plastic bags?
:
Prime factor 96 and 120
96: 2.2.2.2.2.3
120: 2.2.2.3.5
Factors common to both
2.2.2.3 = 24 cookies in each bag
then
2*2 = 4 bags of Nina's cookies
and

Question 990014: Cathy has two strips of colored cardboard with lengths of 72 cm and 84 cm. She wants to cut them into shorter pieces. What is the longest possible length that she can cut both of them so that no cardboard is wasted? How many pieces will she get?
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Cathy has two strips of colored cardboard with lengths of 72 cm and 84 cm. She wants to cut them into shorter pieces. What is the longest possible length that she can cut both of them so that no cardboard is wasted? How many pieces will she get?
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72 = 8*9 = 2^3*3^2
---
84 = 2^2*3*7
----
Greatest common divisor:: 2^2*3 = 12
----
Ans:
longest possible:: 12 cm
# of 12 in 72 = 6 pieces
# of 12 in 84 = 7 pieces
------------
Cheers,
Stan H.
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Question 990019: A dairy farm has two milk loading machines. The two machines started running at the same time. After a few minutes, one machine has loaded 18 cases while the other loaded 24 cases. This is the first time that the two machines finished loading a case simultaneously. What is the longest time possible that the machine could have been running?
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A dairy farm has two milk loading machines. The two machines started running at the same time. After a few minutes, one machine has loaded 18 cases while the other loaded 24 cases. This is the first time that the two machines finished loading a case simultaneously. What is the longest time possible that the machine could have been running?
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18 = 2*3^2
24 = 2^3*3
Least common multiple: = 2^3*3^2 = 8*9 = 72
----
In 72 min the slower machine finishes 72/18 = 4 cases
In 72 min the faster machine finishes 72/24 = 3 cases
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Ans:: 72 minutes
-------
Cheers,
Stan H.
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Question 990018: At the opening of a fair, there were two bands. One had 36 members and the other has 30 members. The band leaders were told to march so that there were as many members marching abreast in a row and both bands must have the same number in a row. If you were one of the band leaders, how would you arrange your band for the march in terms of number of groups and member per group?
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Common divisors of 30 and 36 are:
2, 5, 6, and 10
-----------------
I would arrange the groups as:
30 members:
6 across
5 deep
---------
36 members:
6 across
6 deep

Question 990020: There are 36 teachers and 42 parents playing in sportsfest . What is the greatest number of players per team if a team must have teachers and parents as members ? Each team must have an equal number of teachers and parents as members.
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There are 36 teachers and 42 parents playing in sportsfest . What is the greatest number of players per team if a team must have teachers and parents as members ? Each team must have an equal number of teachers and parents as members.
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Ans: Since there are only 36 teachers, the greatest
# of parents playing must be 36
----
Cheers,
Stan H.
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Question 989719: Show that n(n+1)/(2n+4) is a reducible fraction.

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.
The numerator n*(n+1) is divisible by 2 as the product of two consecutive integers.  So,  the numerator is an even number.

The denominator is  2n + 4 = 2*(n+2)  an even number too.

Thus  they both have the common factor 2.

Question 989462: Find a composite number for each of the following numbers to make each a pair of co-prime numbers?
a: 80, b:63 ,c: 135,d: 88 e:52 f:81 g:256 h:216 i:294 j:1155
Can you please provide answers with the steps to help my daughter explain the same.

Answer by Edwin McCravy(13211)   (Show Source):
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You just have to pick a number that does not share a prime factor with it.
The easiest way to make a composite number with a prime is to square it.

a: 80,

80 has factors 2, 5.  Pick another prime. 3, square it, 9

80 and 9 are coprime.

b:63

63 has prime factors 3, 7.  Pick another prime, 5, square it, 25.

63 and 25 are coprime.

c: 135

135 has prime factors 3,5.  Pick another prime. 7, square it, 49

135 and 49 are coprime.

,d: 88

88 has prime factors 2,11.  Pick another prime. 3, square it, 9.

88 and 9 are coprime

e:52

52 has prime factors 2,13.  Pick another prime, 3, square it, 9

52 and 9 are coprime.

f:81

81 has only the prime factor 3.  Pick another prime 2, square it, 4,

81 and 4 are coprime

g:256

256 has only the prime factor 2.  Pick another prime, 3, square it, 9,

256 and 9 are coprime.

h:216

216 has prime factors 2, 3.  Pick another prime, 5, square it, 25,

216 and 25 are coprime.

i:294

294 has prime factor 2, 3, 7.  Rick another prime, 5. Square it, 25.

294 and 25 are coprime.

j:1155

1155 has prime factors 3, 5, 7, 11. Pick another prime, 2.  Square it, 4.

1155 and 4 are coprime.

Edwin

Question 989049: A rectangle measures 72cm by 108cm.
A 2cm by 2cm square can be used to cover the rectangle without any spaces or overlapping. Explain Why.

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Many, many 2cm by 2cm square can be used to cover the rectangle without any spaces or overlapping.
It is possible because and are both divisible by .
You can start by lining up in a row such 2cm by 2cm squares along one of the long sides,
because .
Then you would add more rows like that for a total of
rows.

Question 988684: The prime factorization of a number is 2 to the power of 5 x3 to the power of8 x5 to the power of 7 x7 to the power of 4x 11x13 Which statements are true about the number? Explain.
a) the number is even
b) number is a multiple of 10
c)15 is a factor of the number
d) 17 is not a factor of the number
e)77 is a factor if the number

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a) True, it has at least one power of 2 making it even.
b) True, it has 2*5 as a factor.
c) True, it has 3*5 as a factor.
d) False, 17 is not a factor.
e) True, it has 7*11 as a factor.

Question 988503: What is 499760 divided by 150
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Divide and find out.

Question 988402: If you add a prime number to itself,is the sum composite or prime? Explain.
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If you add any number to itself, the result is divisible by 2.

John

My calculator said it, I believe it, that settles it

Question 988392: What is the least number that is divisible by 2,3,4,5,7? Explain.
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It has to be even and end in a 0. That will take care of 2,4,and 5.
7X3=21 and their product works for the above if it is 210.
All 5 numbers divide into 210 evenly.
Once you have the first part, you can try 70,140,210.
Alternatively, you can multiply them all together and get 840. That works, then cut it in half. 420 works, and 210 works, but 105 doesn't.

Question 988358: 99s composite factors

Question 985684: Solve using the quadratic equation
-2x² + 5x = 0

Answer by Edwin McCravy(13211)   (Show Source):
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Normally you would not use the quadratic formula to do this
because it is easier to do by factoring.  However when you
are instructed to use a certain method, you must use it.

-2x² + 5x = 0

Write it with a +0 at the end of the left side:

-2x² + 5x + 0 = 0, so it can be compared to
ax² + bx + c = 0

and we can see that a=-2, b=5 and c=0.

Then we write the quadratic formula:

and substitute:

Then simplify:

We use the + sign

So 0 is one solution.

We use the - sign:

So  is the other solution.

Edwin

Question 985369: find the sum of 23, 14, 6
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23+14+6=23+20
=43
result:43

Question 985368: find the sum of 6,9,and 7
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Just add the three numbers . 22

Question 985230:

Question 984736: 10 3/4 ÷ 2 5/6
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Use improper fractions instead of mixed numbers.

Question 983150: A watermelon is cut into two pieces in the ratio 4:5 by weight. Then the larger piece is cut into two pieces in the ratio 7:8 by weight. Out of three pieces what is the ratio of the weight of the largest piece to that of smallest piece?
Found 2 solutions by MathTherapy, josgarithmetic:
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A watermelon is cut into two pieces in the ratio 4:5 by weight. Then the larger piece is cut into two pieces in the ratio 7:8 by weight. Out of three pieces what is the ratio of the weight of the largest piece to that of smallest piece?
Ratio:



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After the first cut, the small part is and the big part is .

Second cut:
The two pieces made from the big part are and .

Attention is now on the fractions ;
;
and
.

Bring the to terms of .
.

Now you are ready to compare.
Which is the largest among ?

The largest is the part which came from the first cutting, which was not cut further.

Question 982773: Hi there
I'm having trouble understanding the answer to this problem:
Question: Convert 318% to a fraction.
I understand 318/100 , but the answer is 3 and 19/100 why is that? How is this the answer? I don't get it please help me understand why!

Found 2 solutions by macston, josgarithmetic:
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.
I think you have a typographical error:
=
The 19/100 should be 18/100 or 318% should be 319%

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Same thing as the other question. Here, you wrote one of the digits wrong.
.

Your other question was asked correctly, and explained in that posting, both by you and by me.

Question 982776: Hello sorry last post was wrong... I'm having trouble understanding the answer to this problem:
Question: Convert 319% to a fraction.
I understand 319/100 , but the answer is 3 and 19/100 why is that? Why is there a 3? from the 319? I don't get it please help me understand why!

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You are showing that you already explained this to yourself.

Question 982672: What is the greatest number of 5 digits which when divided by 16, 24, 30 and 36, 10 will remainder in each case?
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Let  x  be the unknown number.
Then  x-10  is divisible by  16,  24,  30 and  36, according to the condition.

It implies that  x-10  is divisible by  ,    and  .
Hence,  x-10  is divisible by  16*5*9 = 720.

The greatest  5-digits number divided by  720  is  99360  (it is easy to check).
It means that  x-10 = 99360.
Hence,  x = 99370.

Question 982444: Carefully perform the following long division on paper. Be sure you will be able to follow your steps again once you have done the problem.
13 145

Now perform the following long division on paper:
x−3 x2+4x+5

describe what similarities you notice about the two problems?

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The first kind is almost like the second kind in which x=10; but what is different is your binomial divisor SUBTRACTS the lowest x-place, while your base ten numbers uses a divisor in which this divisor ADD the lowest place value count.

Regular long division for base ten numbers condenses the detailed computations, in contrast to how we do polynomial long division; otherwise our regular long division for base ten numbers would be a beg mess.

Question 980566: What does p stand for in this {3x}4
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What does p stand for in this {3x}4
------------
There's no p in that.

Question 979917: Consider all whole numbers from 1 to 9000. How many numbers are divisible by 10 but not by 9? How many numbers are divisible by 2 or 3 (this should include those divisible by both 2 and 3)? Any help would be greatly appreciated.
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9000 divided by 10 is 900, so 900 of the numbers from 1 to 9000 are divisible by 10. 900 divided by 9 is 100, so 100 of the numbers divisible by 10 are also divisible by 9. Therefore 800 of the numbers from 1 to 9000 that are divisible by 10 are NOT divisible by 9.

If something is divisible by 2 AND 3, then it must be divisible by 6.

John

My calculator said it, I believe it, that settles it

Question 978273: if p, q, r, and s are prime numbers and (q^3.p^2)/r^2 = s^n, what is the value of n?
Answer by Edwin McCravy(13211)   (Show Source):
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Prime factorization is unique except for the order of factors.

n could only be 3 because

(q^3*p^2)/r^2 = s^n

then
q^3*p^2= r^2*s^n

There are 5 primes multiplied together on the left, so
there must the same 5 primes multiplied together on the right.

In fact p=r and q=s, and of course n=3.

Edwin

Question 979041: Hi I was wondering for the problem 3/15+2/5, since the GCF of 3 and 15 is 3, how do you get the 1 and 5 out of reducing 3.
Thanks

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You want to raise to higher terms, meaning its equivalent with a larger denominator. Which denominator?

Look at the two denominators which are present:

, and the factor of 1 is understood.
Lowest common denominator is .
Raise the terms of into the equivalent with denominator 15.
-

Question 978614: What is the smallest prime factor of 1001?
Found 2 solutions by Edwin McCravy, Fombitz:
Answer by Edwin McCravy(13211)   (Show Source):
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What is the smallest prime factor of 1001?
The smallest prime is 2.  So we try 2:

500
2)1001
1000
1

So 2 in not a factor of 1001 because it leaves a remainder of 1 when we divide 1001 by 2.

So we try the next prime, which is 3:

333
3)1001
999
2

So 3 in not a factor of 1001 because it leaves a remainder of 2 when we divide 1001 by 3.

So we try the next prime, which is 5

200
5)1001
1000
1

So 5 in not a factor of 1001 because it leaves a remainder of 1 when we divide 1001 by 5.

So we try the next prime, which is 7

143
7)1001
1001
0

So 7 in a factor of 1001 because it leaves a remainder of 0 when we divide 1001 by 7.

Edwin

Question 978612: What is the smallest prime factor of 1001 ?
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Duplicate Question

Question 977715: whats P(2)= 2x^7+5x^2+3
solve with synthetic division and remainder theorem

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whats P(2) if P(x) = 2x^7+5x^2+3
solve with synthetic division and remainder theorem
------
2)....2....0....0.....0.....0.....5.....0.....3
......2....4....8....16....32....69....138...|..279
Ans: P(2) = 279
--------------
Cheers,
Stan H.

Question 977112: show that 3 divides exactly any one of n,n+1 or n+3?
Found 2 solutions by solver91311, Alan3354:
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I think you meant the claim to be that 3 divides one of n, n + 1, or n + 2 since if 3 divides n it must, perforce divide n + 3.

Claim: For all integers n, 3 divides exactly one of n, n + 1, or n + 2

We prove by exhaustive cases using the Generalized Divisibility Theorem

Generalized Divisibility Theorem

such that

and

From which it follows that all integers can be represented by one of the forms:

for for some integer

Cases:

The three cases are exhaustive and each case has the same conclusion, hence the claim is proven.

John

My calculator said it, I believe it, that settles it

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Duplicate, and it doesn't
------------
eg
4, 5 or 7

Question 977111: show that square of every positive integer takes any one of the form 3p or 3p+1?
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A positive integer, when divided by 3, can divide evenly, or have a remainder of 1, or have a remainder of 2.
It must be one of those 3 cases; there is no other possibility.
So, a positive integer can be
a multiple of 3, of the form , or
be of the form (if it has a remainder of 1 when divided by 3), or
be of the form (if it has a remainder of 2 when divided by 3),
with being a non-negative integer in each case.
The square of is with .
The squares of and can be calculated as the square of a binomial with a formula proven in algebra class:
.
The square of is with .
The square of is with .
In other words, when divided by 3, the square of an integer can
divide evenly,
or leave a remainder of 1.
There is no other possibility.

Question 977110: show that cube of each positive integer takes any one of the form 9p, 9p+3,or 9p+8?
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NOT TRUE. Maybe you have a typo.
I could say/prove that the cube of each positive integer takes any one of the forms
, , or ,
with being a non-negative integer,
but the cube of a positive integer cannot take the form with being an integer.

A positive integer, when divided by 3, can divide evenly, or have a remainder of 1, or have a remainder of 2.
It must be one of those 3 cases; there is no other possibility.
So, a positive integer can be
a multiple of 3, of the form , or
be of the form (if it has a remainder of 1 when divided by 3), or
be of the form (if it has a remainder of 2 when divided by 3),
with being a non-negative integer in each case.
The cube of is with .
The cubes of and can be calculated as the cube of a binomial with a formula proven in algebra class:
.
The cube of is with .
The cube of is with .
In other words, when divided by 9, the cube of an integer can
divide evenly,
or leave a remainder of 1,
or leave a remainder of 8.
There is no other possibility.
A number of the form leaves a remainder of 3 when divided by 9, and can never be a perfect cube.

Question 977126: Which of the following is the greatest?
a.
b.
c.
d.
(NOTE: Don't use logarithm. It has been removed from syllabus, and I have to make my students understand without using log)

Found 2 solutions by solver91311, Alan3354:
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We can eliminate a for the reason Alan gave.

Then

By the binomial theorem, and

Notice that in the expansion of , the signs alternate, but in the expansion of the signs are all positive. Therefore

Similarly, the first term of will be and the terms will alternate, hence

Therefore, answer c, as Alan's calculator confirms.

John

My calculator said it, I believe it, that settles it

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Which of the following is the greatest?
a.
b.
c.
d.
---------------
3^210 = 9^105, obviously less than 17^105, so a is eliminated.
-------
Using a calculator:
7^140 = 2.059e118
17^105 = 1.57e129
31^84 = 1.88e125
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c wins again.
======================
You can change them all to the same base, eg, 7, but that would require either logs or a calculator.
----
eg,
7^140 = (2^2.8)^140 = 2^392
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I might have figured that out eventually.
I should have spotted that 3, 7, 17 & 31 are all 2^n plus or minus 1.
In short, his solution is elegant.
Probably with assistance from Satan, since we all know god is stupid.

Question 977085: Which of the following is the greatest?
a.
b.
c.
d.