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SAT Math : Factors / Multiples

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Example Questions

Example Question #11 : How To Factor A Number

If $180 = 2^{a}3^{b}5^{c}7^{d}$ $\displaystyle 180 = 2^{a}3^{b}5^{c}7^{d}$, where $a,b,c,d$ $\displaystyle a,b,c,d$ are all positive integers, what is $a+b+c+d$ $\displaystyle a+b+c+d$?

Possible Answers:

Correct answer:

Explanation:

We will essentially have to represent 180 as a product of prime factors, because 2, 3, 5, and 7 are all prime numbers. The easiest way to do this will be to find the prime factorization of 180.

180 = 18(10)= (9)(2)(10) = (3)(3)(2)(10)=(3)(3)(2)(2)(5) = $2^{2}3^{2}5^{1}$ $\displaystyle 2^{2}3^{2}5^{1}$. Because 7 is not a factor of 180, we can mutiply the prime factorization of 180 by $7^{0}$ $\displaystyle 7^{0}$ (which equals 1) in order to get 7 into our prime factorization.

$180=$ $\displaystyle 180=$ $2^23^25^17^0$ $\displaystyle 2^23^25^17^0$= $2^a3^b5^c7^d$ $\displaystyle 2^a3^b5^c7^d$

In order for $2^23^25^17^0$ $\displaystyle 2^23^25^17^0$ to equal $2^a3^b5^c7^d$ $\displaystyle 2^a3^b5^c7^d$, the exponents of each base must match. This means that a = 2, b = 2, c = 1, and d = 0. The sum of a, b, c, and d is 5.

The answer is 5.

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Example Question #1641 : Sat Mathematics

What is the product of the distinct prime factors of 24?

Possible Answers:

$\dpi{100} \small 24$ $\displaystyle \dpi{100} \small 24$

$\dpi{100} \small 9$ $\displaystyle \dpi{100} \small 9$

$\dpi{100} \small 5$ $\displaystyle \dpi{100} \small 5$

$\dpi{100} \small 8$ $\displaystyle \dpi{100} \small 8$

$\dpi{100} \small 6$ $\displaystyle \dpi{100} \small 6$

Correct answer:

$\dpi{100} \small 6$ $\displaystyle \dpi{100} \small 6$

Explanation:

The prime factorization of 24 is (2)(2)(2)(3). The distinct primes are 2 and 3, the product of which is 6.

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Example Question #11 : How To Factor A Number

How many prime factors does $\dpi{100} \small 2^{3}-1$ $\displaystyle \dpi{100} \small 2^{3}-1$ have?

Possible Answers:

$\dpi{100} \small 5$ $\displaystyle \dpi{100} \small 5$

$\dpi{100} \small 0$ $\displaystyle \dpi{100} \small 0$

$\dpi{100} \small 2$ $\displaystyle \dpi{100} \small 2$

$\dpi{100} \small 3$ $\displaystyle \dpi{100} \small 3$

$\dpi{100} \small 1$ $\displaystyle \dpi{100} \small 1$

Correct answer:

$\dpi{100} \small 1$ $\displaystyle \dpi{100} \small 1$

Explanation:

$\dpi{100} \small 2^{3}-1=8-1=7$ $\displaystyle \dpi{100} \small 2^{3}-1=8-1=7$

Since 7 is prime, its only prime factor is itself.

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Example Question #12 : How To Factor A Number

What is the smallest positive multiple of 12?

Possible Answers:

$\dpi{100} \small 12$ $\displaystyle \dpi{100} \small 12$

$\dpi{100} \small 0$ $\displaystyle \dpi{100} \small 0$

$\dpi{100} \small 2$ $\displaystyle \dpi{100} \small 2$

$\dpi{100} \small 24$ $\displaystyle \dpi{100} \small 24$

$\dpi{100} \small 6$ $\displaystyle \dpi{100} \small 6$

Correct answer:

$\dpi{100} \small 12$ $\displaystyle \dpi{100} \small 12$

Explanation:

Multiples of 12 are found by multiplying 12 by a whole number. Some examples include:

$\dpi{100} \small 12(-2)=-24$ $\displaystyle \dpi{100} \small 12(-2)=-24$

$\dpi{100} \small 12(0)=0$ $\displaystyle \dpi{100} \small 12(0)=0$

$\dpi{100} \small 12(1)=12$ $\displaystyle \dpi{100} \small 12(1)=12$

Clearly, the smallest positive value obtainable is 12. Do not confuse the term multiple with the term factor!

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Example Question #61 : Factors / Multiples

How many prime factors of 210 are greater than 2?

Possible Answers:

four

two

five

three

one

Correct answer:

three

Explanation:

Begin by identifying the prime factors of 210. This can be done easily using a factoring tree (see image).

Vt_p2

The prime factors of 210 are 2, 3, 5 and 7. Of these factors, three of them are greater than 2.

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Example Question #12 : How To Factor A Number

How many integers between 50 and 100 are divisible by 9?

Possible Answers:

Correct answer:

Explanation:

The smallest multiple of 9 within the given range is $\inline \dpi{200} \tiny 54 = 9 \times 6$ $\displaystyle \inline \dpi{200} \tiny 54 = 9 \times 6$.

The largest multiple of 9 within the given range is $\dpi{100} {99=9 \times 11}$ $\displaystyle \dpi{100} {99=9 \times 11}$.

Counting the numbers from 6 to 11, inclusive, yields 6.

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Example Question #11 : Other Factors / Multiples

$\displaystyle N$ is the set of all positive multiples of $\displaystyle 5$, and $\displaystyle W$ is the set of all squares of integers. Which of the following numbers belongs to both sets?

Possible Answers:

1025 $\displaystyle 1025$

225 $\displaystyle 225$

$\displaystyle 20$

150 $\displaystyle 150$

$\displaystyle 75$

Correct answer:

225 $\displaystyle 225$

Explanation:

225 $\displaystyle 225$ is the only choice that is both a multiple of $\displaystyle 5$ and a perfect square.

$5\cdot 45=225$ $\displaystyle 5\cdot 45=225$

$\sqrt{225}=15$ $\displaystyle \sqrt{225}=15$

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SAT Math : Factors / Multiples

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #11 : How To Factor A Number

Example Question #1641 : Sat Mathematics

Example Question #11 : How To Factor A Number

Example Question #12 : How To Factor A Number

Example Question #61 : Factors / Multiples

Example Question #12 : How To Factor A Number

Example Question #11 : Other Factors / Multiples

All SAT Math Resources

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