We need to find a common denominator to add and subtract these fractions. Let's do the addition first. The lowest common denominator of 5 and 7 is 5 * 7 = 35, so 3/5 + 4/7 = 21/35 + 20/35 = 41/35. 

Now to the subtraction. The lowest common denominator of 35 and 3 is 35 * 3 = 105, so altogether, 3/5 + 4/7 – 1/3 = 41/35 – 1/3 = 123/105 – 35/105 = 88/105. This does not simplify and is therefore the correct answer.

Of the four numbers given, 25 is only a factor of 150, since all multiples of 25 end in the digits 25, 50, 75, or 00. To determine whether 150 is correct, we inspect the factors of 150 and 175:

Factors of 150: \(\displaystyle \left \{ 1, 2, 3,5,6,10,15,\underline{25},30, 50, 75, 150\right \}\)

Factors of 175: \(\displaystyle \left \{ 1,5,7,\underline{25}, 35,175\right \}\)

Since 25 is the greatest number in both lists, \(\displaystyle GCF (150,175) = 25\).

The greatest common factor of two numbers is the product of the prime factors they share; if they share no prime factors, it is \(\displaystyle 1\).

(a) \(\displaystyle 16 = 2 \times 2\times 2\times2\). Since \(\displaystyle N\) is an odd prime, \(\displaystyle N\) and \(\displaystyle 16\) share no prime factors, and \(\displaystyle GCF (N,16) = 1\).

(b) \(\displaystyle N^{4} = N \times N \times N \times N\), since \(\displaystyle N\) is prime. Since \(\displaystyle 2\) is an even prime, \(\displaystyle N^{4}\) and \(\displaystyle 2\) share no prime factors, and \(\displaystyle GCF (N^{4},2) = 1\).

The quantities are equal since each is equal to \(\displaystyle 1\).

There are a couple different ways to find the GCF of a set of numbers. Sometimes it's easiest to make a factor tree for each number. The factors that the pair of numbers have in common are then multiplied to get the GCF. So for 45, the prime factorization ends up being: \(\displaystyle 5\cdot 3\cdot 3\). The prime factorization of 120 is: \(\displaystyle 5\cdot 3\cdot 2\cdot 2\cdot 2\). Since they have a 5 and 3 in common, those are multiplied together to get 15 for the GCF. Repeat the same process for 38 and 114. The prime factorization of 38 is \(\displaystyle 19\cdot 2\). The prime factorization of 114 is \(\displaystyle 19\cdot 3\cdot 2\). Therefore, multiply 19 and 2 to get 38 for their GCF. Column B is greater.

To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:

\(\displaystyle 630 = 63 * 10 = 7 * 9 * 2 * 5 = 7 * 3 * 3 * 2 * 5\)

        \(\displaystyle = 2 * 3^2* 5*7\)

\(\displaystyle 660 = 66 * 10 = 6 * 11 * 2 * 5 = 2 * 3 * 11 * 2 *5\)

        \(\displaystyle = 2^2 *3*5*11\) 

Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:

\(\displaystyle 2\): \(\displaystyle 2\)

\(\displaystyle 3\): \(\displaystyle 3\)

\(\displaystyle 5\): \(\displaystyle 5\)

\(\displaystyle 7\): None

\(\displaystyle 11\):  None

Taking these together, you get:

\(\displaystyle 2*3*5 = 30\)

To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:

\(\displaystyle 520 = 52 * 10 = 13 * 4 * 2 * 5 = 13 * 2 * 2* 2 *5\)

        \(\displaystyle = 2^3 *5*13\)

\(\displaystyle 175 = 5 * 35 = 5 * 5 * 7 = 5^2 * 7\) 

Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:

\(\displaystyle 2\): None

\(\displaystyle 5\): \(\displaystyle 5\)

\(\displaystyle 7\): None

\(\displaystyle 13\): None

Taking these together, you get: \(\displaystyle 5\)

To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:

\(\displaystyle 208 = 4 * 52 = 2 * 2 * 4*13 = 2 * 2 * 2 * 2* 13\)

        \(\displaystyle =2^4 * 13\)

\(\displaystyle 204 = 6 * 34 = 2 * 3 * 2 * 17 = 2^2 * 3 * 17\) 

Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:

\(\displaystyle 2\): \(\displaystyle 2^2\)

\(\displaystyle 3\): None

\(\displaystyle 13\): None

\(\displaystyle 17\): None

Taking these together, you get: \(\displaystyle 2^2 = 4\)

If Annette's family has \(\displaystyle 8\frac{3}{5}\) jars of applesauce, and in a month, they go through \(\displaystyle 4\frac{2}{3}\) jars of apple sauce, that means \(\displaystyle 8\frac{3}{5}-4\frac{2}{3}\) jars of applesauce will be left. 

The first step to determining how much applesauce is left it to convert the fractions into mixed numbers. This gives us:

\(\displaystyle \frac{43}{5}-\frac{14}{3}\)

The next step is to find a common denominator, which would be 15. This gives us:

\(\displaystyle \frac{129}{15}-\frac{70}{15}\)

\(\displaystyle \frac{59}{15}\)

\(\displaystyle 3\frac{14}{15}\)

If two integers are broken down into their prime factorizations, their greatest common factor is the product of their common prime factors.

Since \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), \(\displaystyle d\), and \(\displaystyle e\) are distinct prime integers, the two expressions can be factored into their prime factorizations as follows - with their common prime factors underlined:

\(\displaystyle abc^{2}d = \underline{a} \cdot \underline{b} \cdot c \cdot c \cdot \underline{d}\)

\(\displaystyle a^{2}bde^{2} = \underline{a} \cdot a \cdot \underline{b} \cdot \underline{d} \cdot e \cdot e\)

The greatest common factor is the product of those three factors, or \(\displaystyle abd\).

For two numbers to be relatively prime, they cannot have any factor in common except for 1. The factors of 18 are 1, 2, 3, 6, 9, and 18.

We can eliminate 32 and 34, since each shares with 18 a factor of 2; we can also eliminate 33 and 39, since each shares with 18 a factor of 3. The factors of 35 are 1,  5, 7, and 35; as can be seen by comparing factors, 18 and 35 only have 1 as a factor, making 35 the correct choice.