Divisibility & Number Fluency - SAT Math

Card 0 of 124

Question

The first 3 prime numbers multiplied together equal:

Answer

The correct answer is 30. A prime number is any number that is only divisible by 1 and itself. 0 and 1 are not prime numbers, so the first 3 prime numbers are 2, 3, and 5.

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Question

Which of the following numbers is not divisible by 3:

Answer

A common rule about numbers divisible by 3 is that the sum of their digits should be divisible by 3. 4+1+2 = 7 and 7 is not divisible by 3. Therefore, 412 is not divisible by 3.

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Question

If and , which of the following must be true?

Answer

Since but there are no individual statements about each number on its own, we can’t conclude if a, b, c, or d are positive or negative. Recall that two negative numbers will give a positive product. What we can conclude, though, is that neither a, b, c, or d can equal 0 because then the product would equal 0. This means . A could be greater than 0 or could be less than 0 if either b, c, or d were negative. C could be less than 1, but C could also be a positive number if b or e are negative OR if the product of a, b, and d is positive. Similarly, d could be positive and would support the positive product of 10.

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Question

Which of the following are real numbers?

I)

II) $\sqrt{5}$

III)

Answer

Complex numbers (any number in the form of ) are not real numbers. $\sqrt{5}$ is irrational, but rational and irrational numbers both fall into the category of real numbers. is an integer, which is a subcategory of real numbers.

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Question

Which of the following are rational numbers?

I)

II) $\pi$

III)

Answer

$\pi$ (pi) is an irrational number. Any square roots or fractions that result in non-terminating, non-repeating decimals fall under the category of irrational numbers. and are both terminating decimals and thus fall under the category of rational numbers.

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Question

Which of the following are whole numbers?

I)

II) $\sqrt{4}$

III)

Answer

Negative rational numbers that don’t have decimals are integers. Whole numbers are a subcategory of integers that only include positive numbers and 0. At first glance, it seems that any square root would fall into irrational numbers, but this particular radical can be simplified to 2, making it a whole number.

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Question

The correct prime factorization of 210 is?

Answer

Recall that 1 is not a prime number. A prime number is any number greater than 1 that can only be divided by 1 and itself. All the above options are factors of 210, but neither 6 nor 10 are prime numbers (6 can be divided into 2 and 3 while 10 can be divided into 2 and 5).

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Question

What is the least common multiple of 7, 6, and 10?

Answer

The least common multiple (LCM) refers to the smallest multiple for each number that is shared by all the numbers of a given set, so in this case, it is the smallest multiple of 7, 6, and 10. The best way to approach these problems is to factorize each number and calculate the LCM from the frequency of these numbers. 7 is a prime number, 6 can be factorized into 2 and 3, and 10 can be factorized into 2 and 5. Now, we multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs. LCM = 7 * 2 * 3 * 5 = 210. We do not count 2 twice.

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Question

What is the least common multiple of 30 and 45?

Answer

The least common multiple (LCM) refers to the smallest multiple for each number that is shared by all the numbers of a given set, so in this case, it is the smallest multiple of 30 and 45. The best way to approach these problems is to factorize each number and calculate the LCM from the frequency of these numbers. 30 can be factorized into 2, 3 and 5. 45 can be factorized into 3, 3, and 5. Now, we multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers (like it does in 45), you multiply the factor the greatest number of times it occurs. LCM = 2 * 3 * 3 * 5 = 90.

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Question

What is the greatest common factor of 15 and 75?

Answer

The greatest common factor (GCF) refers to the largest factor for each number that is shared by all the numbers of a given set, so in this case, it is the greatest factor of 15 and 75. The best way to approach these problems is to factorize each number and calculate the GCF from the multiplication of these numbers. 15 can be factorized into 3 and 5. 75 can be factorized into 3, 5, and 5. Now, identify the factors they have in common. Both 15 and 75 share one 3 and one 5, so multiplied together, the GCF = 3 * 5 = 15.

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Question

Simplify $\frac{x}{2}-\frac{x}{5}$

Answer

Simplifying this expression is similar to $\frac{1}{2}-\frac{1}{5}$ . The denominators are relatively prime (have no common factors) so the least common denominator (LCD) is . So the problem becomes $\frac{1}{2}-\frac{1}{5}=\frac{5}{10}-\frac{2}{10}=\frac{3}{10}$ .

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Question

If $\frac{p}{6}$ is an integer, which of the following is a possible value of ?

Answer

$\frac{0}{6}=0$ , which is an integer (a number with no fraction or decimal part). All the other choices reduce to non-integers.

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Question

Simplify: $\frac{4x^{5}y^{3}z}{12x^{3}y^{6}z^{2}}$

Answer

$\frac{4x^{5}y^{3}z}{12x^{3}y^{6}z^{2}}=\frac{x^{2}}{3y^{3}z}$

First, let's simplify $\frac{4}{12}$ . The greatest common factor of 4 and 12 is 4. 4 divided by 4 is 1 and 12 divided by 4 is 3. Therefore $\frac{4}{12}=\frac{1}{3}$ .

To simply fractions with exponents, subtract the exponent in the numerator from the exponent in the denominator. That leaves us with $\frac{1}{3}x^{2}y^{-3}z^{-1}$ or $\frac{x^{2}}{3y^{3}z}$

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Question

Which of the following is not equal to $\frac{32}{24}$ ?

Answer

$\frac{32}{24}=1.33$ , so the answer choice needs to be something that is not equal to in order to be correct.

$\frac{16}{12}=1.33$

$\frac{224}{168}=1.33$

$\frac{4}{3}=1.33$

$\frac{160}{96}=1.67$

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Question

Find the root of:

$y=\frac{x^2-\sqrt3x+\sqrt5x-\sqrt{15}}{x-\sqrt3}$

Answer

$y=\frac{x^2-\sqrt3x+\sqrt5x-\sqrt{15}}{x-\sqrt3}=\frac{(x-\sqrt3)(x+\sqrt5)}{x-\sqrt3}=x+\sqrt5$

The root occurs where . So we substitute 0 for .

$y=x+\sqrt{5}$

$0=x+\sqrt{5}$

This means that the root is at $x=-\sqrt5$ .

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Question

Simplify $\frac{32}{88}$

Answer

Find the common factors of the numerator and denominator. They both share factors of 2,4, and 8. For simplicity, factor out an 8 from both terms and simplify.

$\frac{32}{88}=\frac{8\times 4}{8\times 11}=\frac{4}{11}$

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Question

Simply the following fraction:

$\frac{\frac{a^2}{b}}{\frac{a}{b}}$

Answer

Remember that when you divide a fraction by a fraction, that is the same as multiplying the fraction in the numerator by the reciprocal of the fraction in the denominator.

In other words,

$\frac{\frac{a^2}{b}}{\frac{a}{b}} = \frac{a^2}{b}\cdot\frac{b}{a}=\frac{a^2b}{ab}$

Simplifying this final fraction gives us our correct answer, .

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Question

Solve for .

$\frac{\frac{x-2}{4^2}}{\frac{x^2-4}{2}}=1$

Answer

To solve for , simplify the fraction. In order to do this, recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, rewrite the equation as follows.

$\frac{\frac{x-2}{4^2}}{\frac{x^2-4}{2}}=1\Rightarrow \frac{x-2}{4^2}\times \frac{2}{x^2-4}=1$

Now, simplify the first fraction by calculating four squared.

$\frac{x-2}{16}\times \frac{2}{x^2-4}=1$

From here, factor the denominator of the second fraction.

$\frac{x-2}{16}\times \frac{2}{(x-2)(x+2)}=1$

Next, factor the 16.

$\frac{x-2}{2(8)}\times \frac{2}{(x-2)(x+2)}=1$

From here, cancel out like terms that are in both the numerator and denominator. In this particular case that includes (x-2) and 2.

$\frac{1}{8(x+2)}=1$

Now, distribute the eight.

$\frac{1}{8x+16}=1$

Next, multiply both sides by the denominator.

$(8x+16)\times \frac{1}{8x+16}=1\times (8x+16)$

The (8x+16) cancels out and leaves the following equation.

Now to solve for perform opposite operations to move all numerical values to one side of the equation leaving by itself on the other side of the equation.

$\1=8x+16 \-15=8x \\\frac{-15}{8}=x$

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Question

Which of the following fractions is not equivalent to $\frac{6}{45}$ ?

Answer

Let us simplify $\frac{6}{45}$ :

$\frac{6}{45}=\frac{3\times 2}{3\times 15}=\frac{2}{15}$

We can get alternate forms of the same fraction by multiplying the denominator and the numerator by the same number:

$\frac{2\times 2}{15\times 2}=\frac{4}{30}$

$\frac{2\times 1.5}{15\times 1.5}=\frac{3}{22.5}$

Now let's look at $\frac{12}{89}$ :

$6 \times 2 = 12$ , but $45 \times 2 = 90$ .

Therefore, $\frac{12}{89}$ is the correct answer, as it is not equivalent to $\frac{6}{45}$ .

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Question

Which of the following fractions has the greatest value?

Answer

To best compare these fractions, it is helpful to reduce them to their simplest form. $\frac{7}{6}$ is already as reduced as it can be.

For $\frac{152}{132}$ , you can factor out a 2 from the numerator and denominator to get to $\frac{76}{66}$ . Note that you could again divide by 2 to get $\frac{38}{33}$ , but here you can also be creative: if you were comparing to $\frac{7}{6}$ and $\frac{77}{65}$ as two options, you should see that $\frac{76}{66}$ is a hair less than $\frac{77}{66}$ , which would reduce directly to $\frac{7}{6}$ . So you can eliminate $\frac{152}{132}$ by that comparison.

7007 and 6006 are each divisible by 1001 (something that should be somewhat evident by the similar way the numbers look). If you then factor out the 1001 you have $\frac{7(1001)}{6(1001)}$ and the 1001s cancel, leaving $\frac{7}{6}$ . Since you cannot have two correct answers, then $\frac{7007}{6006}$ and $\frac{7}{6}$ are a "tie" and both are incorrect. And when you see that $\frac{77}{65}$ is just a bit larger than $\frac{77}{66}$ , a number that would also equal $\frac{7}{6}$ , you can see that $\frac{77}{65}$ is correct.

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