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ISEE Upper Level Quantitative : Numbers and Operations

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #4 : Least Common Multiple

Multiply the least common multiple of 504 and 624 by the greatest common factor of 504 and 624.

Possible Answers:

104,832

157,248

52,416

628,992

314,496

Correct answer:

314,496

Explanation:

The product of the least common multiple of any two integers and the greatest common factor of the same two integers is the product of the two integers themselves. Therefore,

$LCD (504, 624) \times GCF (504, 624) = 504 \times 624 = 314,496$ .

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Example Question #7 : How To Find The Least Common Multiple

, , , , and are five distinct prime integers. Give the least common multiple of $abc^{2}d$ and $a^{2}bde^{2}$ .

Possible Answers:

abcde

$a^{2}bd$

$a^{2}b^{2}c^{2}d^{2}e^{2}$

abd

$a^{2}bc^{2}de^{2}$

Correct answer:

$a^{2}bc^{2}de^{2}$

Explanation:

If two integers are broken down into their prime factorizations, their greatest common factor is the product of the prime factors that appear in one or both factorizations.

Since , , , , and are distinct prime integers, the two expressions can be factored into their prime factorizations as follows - with their common prime factors underlined:

$abc^{2}d \; \; = a\; \; \; \; \cdot b \cdot c \cdot c \cdot d$

$a^{2}bde^{2} = a \cdot a \cdot b \; \; \; \;\; \; \; \; \cdot d \cdot e \cdot e$

The LCM collects each of the factors:

$LCM = a \cdot a \cdot b \cdot c \cdot c \cdot d \cdot e \cdot e$

$= a^{2}bc^{2}de^{2}$

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Example Question #8 : How To Find The Least Common Multiple

Define an operation $\bigstar$ as follows:

For all positive integers and

$a \bigstar b = \textup{lcd }(a,b) + \textup{gcf }(a,b)$ .

Evaluate $100 \bigstar 80$ .

Possible Answers:

516

1,008

180

240

420

Correct answer:

420

Explanation:

To find the LCD and GCF of 100 and 80, first, find their prime factorizations:

$80 =\underline{2} \cdot\underline{ 2} \cdot 2 \cdot 2 \cdot \underline{5}$

$100 = \underline{2} \cdot \underline{2} \cdot \underline{5} \cdot 5$

The GCF of the two is the product of their shared prime factors, so

$\textup{gcf }(100, 80) = 2 \cdot 2 \cdot 5 = 20$

The LCM is the product of all factors that occur in one or the other factorization, so

$\textup{lcm }(100, 80) = 2 \cdot 2\cdot 2\cdot 2 \cdot 5 \cdot 5 = 400$

Add:

$a \bigstar b = \textup{lcd }(a,b) + \textup{gcf }(a,b) = 400 + 20 = 420$

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Example Question #6 : How To Find The Least Common Multiple

Which of the following is the least common multiple of and 40?

Possible Answers:

160

250

150

200

Correct answer:

200

Explanation:

List the first few multiples of both and $40\textup:$

$25: 25, 50, 75, 100, 125, 150, 175, \underline{200},...$

$40: 40, 80, 120, 160, \underline{200}, 240,...$

The least number in both lists of factors is 200 .

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Example Question #101 : Numbers And Operations

Simplify: $\left (y ^{3} \right )^{4}$

Possible Answers:

$y ^{81}$

The correct answer is not among the other choices.

$y ^{12}$

$y ^{64}$

$y ^{7}$

Correct answer:

$y ^{12}$

Explanation:

Apply the power of a power property:

$\left (y ^{3} \right )^{4} = y ^{3 \cdot 4} = y^{12}$

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Example Question #2 : How To Multiply Exponents

Simplify the expression: $\left (5 a^{3}y ^{4} \right )^{2}$

Possible Answers:

$10a^{5} y^{6}$

$25a^{6} y^{8}$

$10a^{6} y^{8}$

$10a^{9} y^{16}$

$25a^{9} y^{16}$

Correct answer:

$25a^{6} y^{8}$

Explanation:

Apply the power of a product rule, then apply the power of a power rule:

$\left (5 a^{3}y ^{4} \right )^{2}$

$= 5^{2} \cdot \left (a^{3}\right )^{2} \cdot \left ( y ^{4} \right )^{2}$

$= 25 \cdot a^{ 3 \cdot 2 } \cdot y ^{ 4 \cdot 2 }$

$= 25 a^{ 6} y ^{ 8 }$

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Example Question #3 : How To Multiply Exponents

Which of the following expressions is equal to $1,234^{0}$ ?

Possible Answers:

1,234

The expression is undefined.

$\frac{1}{1,234 }$

Correct answer:

Explanation:

Any nonzero number raised to the power of 0 is equal to 1.

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Example Question #102 : Numbers And Operations

Which quantity is greater?

(a) $(-5) ^{3}$

(b) $(-5) ^{-3}$

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

(a) $(-5) ^{3} = (-5) \cdot (-5) \cdot (-5) = 25 \cdot (-5) = -125$

(b) $(-5) ^{-3} = \frac{1}{(-5) ^{3}} = \frac{1}{-125} =- \frac{1}{125}$

(b) is the greater quantity.

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Example Question #1 : How To Multiply Exponents

is positive.

Which is the greater quantity?

(a) $100 ^{2x}$

(b) $1,000 ^{x}$

Possible Answers:

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

Correct answer:

(a) is greater

Explanation:

Use the power of a power property:

(a) $100 ^{2x} = \left (10^{2} \right ) ^{2x} = 10^{2\cdot 2x }= 10^{4x }$

(b) $1,000 ^{x} =\left ( 1 0 ^{3} \right ) ^{x} = 1 0 ^{3x}$

Since x > 0 , 4x > 3x . Subsequently,

$100 ^{2x} = 10^{4x } > 10^{3x } = 1,000^{x }$ ,

making (a) greater

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Example Question #5 : How To Multiply Exponents

Which is the greater quantity?

(a) $2 ^{100}$

(b) $4^{50}$

Possible Answers:

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

$4^{50} = \left ( 2 ^{2} \right )^{50} = 2 ^{ 2\; \cdot \;50 } = 2 ^{ 100 }$

The two quantities are equal.

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

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ISEE Upper Level Quantitative : Numbers and Operations

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #4 : Least Common Multiple

Example Question #7 : How To Find The Least Common Multiple

Example Question #8 : How To Find The Least Common Multiple

Example Question #6 : How To Find The Least Common Multiple

Example Question #101 : Numbers And Operations

Example Question #2 : How To Multiply Exponents

Example Question #3 : How To Multiply Exponents

Example Question #102 : Numbers And Operations

Example Question #1 : How To Multiply Exponents

Example Question #5 : How To Multiply Exponents

All ISEE Upper Level Quantitative Resources

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