Number Theory

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SAT Math › Number Theory

Questions 1 - 10

Sets

The above represents a Venn diagram. The universal set is the set of all positive integers.

Let be the set of all multiples of 2; let be the set of all multiples of 3; let be the set of all multiples of 5.

As you can see, the three sets divide the universal set into eight regions. Suppose each positive integer was placed in the correct region. Which of the following numbers would be in the same region as 873?

Explanation

From the last digit, it can be immediately determined that 873 is not a multple of 2 or 5; since $873 \div 3 = 291$ , 873 is a multiple of 3. Therefore,

$873 \in A' \cap B \cap C'$

We are looking for an integer that is also in this set - that is, one that is also a multiple of 3 but not 2 or 5. From the last digits, we can immediately eliminate 366 and 368 as multiples of 2 and 365 as a multiple of 5. We test 367 and 369 to see which one is a multiple of 3:

$367 \div 3 = 122 \textrm{ R }1$

$369 \div 3 = 123$

369 is the correct choice.

Sets

The above represents a Venn diagram. The universal set is the set of all positive integers.

Let be the set of all multiples of 2; let be the set of all multiples of 3; let be the set of all multiples of 5.

Explanation

From the last digit, it can be immediately determined that 873 is not a multple of 2 or 5; since $873 \div 3 = 291$ , 873 is a multiple of 3. Therefore,

$873 \in A' \cap B \cap C'$

$367 \div 3 = 122 \textrm{ R }1$

$369 \div 3 = 123$

369 is the correct choice.

Multiply:

Explanation

Use the FOIL technique:

$= 7 \cdot 1 - 7 \cdot 2i + 3i \cdot 1 - 3i \cdot 2i$

$= 7 - 14i + 3i - 6i ^{2}$

Which answer choice has the greatest real number value?

$i^4 + i^8 + i^{12} +i^ {16}$

$i^2 + i^8 + i^{10} +i^ {16}$

$i^4 + i^6 + i^{14} +i^ {24}$

$i^{40} + i^2 + i^{13} +i^ {11}$

$i + i^3 + i^{7} +i^ {9}$

Explanation

Recall the definition of and its exponents

$i = \sqrt[]{-1}$

because then

$i^5 = i^4\cdot i =1\cdot i =i$ .

We can generalize this to say

$i^n =i^{n-4}$

Any time is a multiple of 4 then . For any other value of we get a smaller value.

For the correct answer each of the terms equal

$i^4 =i^8 =i^{12} = i^{16} =1$

So:

$i^4 + i^8 + i^{12} +i^ {16}$

Because all the alternative answer choices have 4 terms, and each answer choice has at least one term that is not equal to they must all be less than the correct answer.

Multiply:

Explanation

Use the FOIL technique:

$= 7 \cdot 1 - 7 \cdot 2i + 3i \cdot 1 - 3i \cdot 2i$

$= 7 - 14i + 3i - 6i ^{2}$

Evaluate:

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

Explanation

We can set in the cube of a binomial pattern:

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

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

$= 2^{3} + 3 \cdot 2^{2} \cdot 3i + 3 \cdot 2 \cdot 3 ^{2}\cdot i^{2} + 3^{3} \cdot i^{3}$

$= 8 + 36i + 54 i^{2} + 27 i^{3}$

Evaluate:

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

Explanation

We can set in the cube of a binomial pattern:

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

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

$= 2^{3} + 3 \cdot 2^{2} \cdot 3i + 3 \cdot 2 \cdot 3 ^{2}\cdot i^{2} + 3^{3} \cdot i^{3}$

$= 8 + 36i + 54 i^{2} + 27 i^{3}$

Which answer choice has the greatest real number value?

$i^4 + i^8 + i^{12} +i^ {16}$

$i^2 + i^8 + i^{10} +i^ {16}$

$i^4 + i^6 + i^{14} +i^ {24}$

$i^{40} + i^2 + i^{13} +i^ {11}$

$i + i^3 + i^{7} +i^ {9}$

Explanation

Recall the definition of and its exponents

$i = \sqrt[]{-1}$

because then

$i^5 = i^4\cdot i =1\cdot i =i$ .

We can generalize this to say

$i^n =i^{n-4}$

Any time is a multiple of 4 then . For any other value of we get a smaller value.

For the correct answer each of the terms equal

$i^4 =i^8 =i^{12} = i^{16} =1$

So:

$i^4 + i^8 + i^{12} +i^ {16}$

Because all the alternative answer choices have 4 terms, and each answer choice has at least one term that is not equal to they must all be less than the correct answer.

Multiply:

$-3 i \cdot 8i \cdot 5i \cdot 6i$