SAT II Math I : Number Theory

Study concepts, example questions & explanations for SAT II Math I

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Number Theory

Express 374 in base six.

Possible Answers:

\(\displaystyle 1422_{\textrm{six}}\)

\(\displaystyle 1402_{\textrm{six}}\)

\(\displaystyle 1502_{\textrm{six}}\)

\(\displaystyle 1522_{\textrm{six}}\)

Correct answer:

\(\displaystyle 1422_{\textrm{six}}\)

Explanation:

To convert a base ten number to base six, divide the number by six, with the remainder being the digit in the units place; continue, dividing each successive quotient by 6 and putting the remainder in the next position to the left until the final quotient is less than 6.

\(\displaystyle 374 \div 6 = 62\textrm{ R }\underline{2}\) - the rightmost digit is 2

\(\displaystyle 62 \div 6 = 10 \textrm{ R }\underline{2}\) - the next digit to the left is 2

\(\displaystyle 10 \div 6 = \underline{1} \textrm{ R }\underline{4}\) - the next digit to the left is 4, and the digit to the left of that is 1.

\(\displaystyle 374 = 1422_{\textrm{six}}\)

Example Question #1 : Types Of Numbers

Express  \(\displaystyle 13241 _{\textrm{five}}\) in base ten.

Possible Answers:

\(\displaystyle 921\)

\(\displaystyle 1,121\)

\(\displaystyle 971\)

\(\displaystyle 1,021\)

\(\displaystyle 1,071\)

Correct answer:

\(\displaystyle 1,071\)

Explanation:

Place values in the base five system are powers of five rather than powers of ten. \(\displaystyle 13241 _{\textrm{five}}\) is equal to

\(\displaystyle 1 \times 5^{4} + 3 \times 5 ^{3} + 2 \times 5^{2} + 4 \times 5 + 1\)

\(\displaystyle = 1 \times 625 + 3 \times 125 + 2 \times 25 + 4 \times 5 + 1\)

\(\displaystyle = 625 +375 + 50 + 20 + 1 = 1,071\)

Example Question #1 : Sat Subject Test In Math I

Which of the following is the smallest prime number:\(\displaystyle 8,9,11,13,17\)?

Possible Answers:

\(\displaystyle 11\)

\(\displaystyle 9\)

\(\displaystyle 8\)

\(\displaystyle 13\)

\(\displaystyle 17\)

Correct answer:

\(\displaystyle 11\)

Explanation:

A prime number has only two factor, \(\displaystyle 1\) and itself.  

This is not true for \(\displaystyle 8 (2*4=8)\) or \(\displaystyle 9 (3*3*=9)\) so they are not the answer.  

\(\displaystyle 11, 13, 17\) are all prime numbers and \(\displaystyle 11\) is the smallest so it is your answer.

Example Question #1 : Sat Subject Test In Math I

Which of the following is a rational number?

\(\displaystyle \frac{1}{2},\sqrt{5},2i,3.658\)

Possible Answers:

\(\displaystyle 3.658\)

\(\displaystyle 2i\)

None

\(\displaystyle \sqrt{5}\)

\(\displaystyle \frac{1}{2}\)

Correct answer:

\(\displaystyle \frac{1}{2}\)

Explanation:

A rational number is any number that can be expressed as an integer p or a fraction p/q. In the set of number given only \(\displaystyle \tiny \frac{1}{2}\) fits into that category.

Example Question #1 : Irrational Numbers

Which of the following is not an irrational number?

Possible Answers:

\(\displaystyle \sqrt[3]{125}\)

\(\displaystyle \sqrt[5]{125}\)

\(\displaystyle \sqrt[2]{125}\)

\(\displaystyle \sqrt[4]{125}\)

\(\displaystyle \sqrt[6]{125}\)

Correct answer:

\(\displaystyle \sqrt[3]{125}\)

Explanation:

A root of an integer is one of two things, an integer or an irrational number. By testing all five on a calculator, only \(\displaystyle \sqrt[3]{125}\) comes up an exact integer - 5. This is the correct choice.

Example Question #1 : Irrational Numbers

Simplify by rationalizing the denominator:

\(\displaystyle \frac{45}{5 + \sqrt{7}}\)

Possible Answers:

\(\displaystyle \frac{ 25 + \sqrt{7} }{ 2}\)

\(\displaystyle \frac{ 25 - 5 \sqrt{7} }{ 2}\)

\(\displaystyle \frac{ 25 + 5 \sqrt{7} }{ 2}\)

\(\displaystyle \frac{ 25 - \sqrt{7} }{ 2}\)

\(\displaystyle \frac{15 \sqrt{7}}{4}\)

Correct answer:

\(\displaystyle \frac{ 25 - 5 \sqrt{7} }{ 2}\)

Explanation:

Multiply the numerator and the denominator by the conjugate of the denominator, which is \(\displaystyle 5 - \sqrt{7}\). Then take advantage of the distributive properties and the difference of squares pattern:

\(\displaystyle \frac{45}{5 + \sqrt{7}}\)

\(\displaystyle = \frac{45\left (5 - \sqrt{7} \right )}{\left (5 + \sqrt{7} \right )\left (5 - \sqrt{7} \right )}\)

\(\displaystyle = \frac{45\left (5 - \sqrt{7} \right )}{ 5 ^{2}- \left ( \sqrt{7} \right ) ^{2} }\)

\(\displaystyle = \frac{45\left (5 - \sqrt{7} \right )}{ 25 -7}\)

\(\displaystyle = \frac{45\left (5 - \sqrt{7} \right )}{ 18}\)

\(\displaystyle = \frac{5\left (5 - \sqrt{7} \right )}{ 2}\)

\(\displaystyle = \frac{ 25 - 5 \sqrt{7} }{ 2}\)

Example Question #7 : Number Theory

\(\displaystyle \frac{-5+10i}{3+4i}=?\)

Possible Answers:

\(\displaystyle -1+2i\)

\(\displaystyle -\frac{5}{3}+\frac{5}{2}i\)

\(\displaystyle -\frac{5}{3}-\frac{5}{2}i\)

\(\displaystyle 1-2i\)

\(\displaystyle 1+2i\)

Correct answer:

\(\displaystyle 1+2i\)

Explanation:

\(\displaystyle \frac{-5+10i}{3+4i}\)

\(\displaystyle =\frac{(-5+10i)(3-4i)}{(3+4i)(3-4i)}\)

\(\displaystyle =\frac{-15+20i+30i-40i^{2}}{9-16i^{2}}\)

\(\displaystyle =\frac{-15+50i+40}{9+16}\)

\(\displaystyle =\frac{25+50i}{25}\)

\(\displaystyle =1+2i\)

Example Question #8 : Number Theory

Multiply:

\(\displaystyle (7 + 3i) (1 - 2i)\)

Possible Answers:

\(\displaystyle 13 - 17i\)

\(\displaystyle 7 - 17i\)

\(\displaystyle 7 - 5i\)

\(\displaystyle 1 - 11i\)

\(\displaystyle 13 - 11i\)

Correct answer:

\(\displaystyle 13 - 11i\)

Explanation:

Use the FOIL technique:

\(\displaystyle (7 + 3i) (1 - 2i)\)

\(\displaystyle = 7 \cdot 1 - 7 \cdot 2i + 3i \cdot 1 - 3i \cdot 2i\)

\(\displaystyle = 7 - 14i + 3i - 6i ^{2}\)

\(\displaystyle = 7 - 14i + 3i - 6(-1)\)

\(\displaystyle = 7 - 14i + 3i +6\)

\(\displaystyle = 13 - 11i\)

Example Question #1 : Irrational Numbers

Evaluate: 

\(\displaystyle \left (2 + 3i \right )^{3}\)

Possible Answers:

\(\displaystyle 8 + 9i\)

\(\displaystyle 8 - 27i\)

\(\displaystyle -4 6+ 9i\)

\(\displaystyle -46 + 63 i\)

\(\displaystyle -62 + 63 i\)

Correct answer:

\(\displaystyle -4 6+ 9i\)

Explanation:

We can set \(\displaystyle A = 2, B = 3i\) in the cube of a binomial pattern:

\(\displaystyle \left ( A + B\right ) ^{3} = A ^{3} + 3 A^{2 }B + 3AB^{2} + B^{3}\)

\(\displaystyle = 2 ^{3} + 3 \cdot 2 ^{2 } \cdot 3i + 3 \cdot 2 \cdot (3i)^{2} + (3i)^{3}\)

\(\displaystyle = 2^{3} + 3 \cdot 2^{2} \cdot 3i + 3 \cdot 2 \cdot 3 ^{2}\cdot i^{2} + 3^{3} \cdot i^{3}\)

\(\displaystyle = 8 + 36i + 54 i^{2} + 27 i^{3}\)

\(\displaystyle = 8 + 36i + 54 (-1) + 27 (-i)\)

\(\displaystyle = 8 + 36i - 54 - 27 i\)

\(\displaystyle = -46 + 9i\)

Example Question #2 : Irrational Numbers

Evaluate \(\displaystyle \small \frac{-6+2i}{10-3i}\)

Possible Answers:

\(\displaystyle \small \frac{-54+38i}{109}\)

\(\displaystyle \small \small \frac{-54+38i}{91-60i}\)

\(\displaystyle \small \frac{-66+2i}{109}\)

\(\displaystyle \small \small \frac{-66+2i}{91-60i}\)

You cannot divide by complex numbers

Correct answer:

\(\displaystyle \small \frac{-66+2i}{109}\)

Explanation:

To divide by a complex number, we must transform the expression by multiplying it by the complex conjugate of the denominator over itself. In the problem, \(\displaystyle \small 10-3i\) is our denominator, so we will multiply the expression by \(\displaystyle \small \frac{10+3i}{10+3i}\) to obtain:

\(\displaystyle \small \frac{-6+2i}{10-3i}\frac{10+3i}{10+3i}=\frac{-60+20i-18i+6i^2}{100-30i+30i-9i^2}\).

We can then combine like terms and rewrite all \(\displaystyle \small i^2\) terms as \(\displaystyle \small -1\). Therefore, the expression becomes:

\(\displaystyle \small \frac{-60+2i+6i^2}{100-9i^2}=\frac{-66+2i}{109}\)

Our final answer is therefore \(\displaystyle \small \frac{-66+2i}{109}\)

Learning Tools by Varsity Tutors