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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

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Example Questions

Example Question #33 : How To Find Out If A Number Is Prime

and are prime numbers.

80 < M < 90 <N < 100

Which is the greater quantity?

(A) M + N

(B) 175

Possible Answers:

(B) is greater

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(A) is greater

Correct answer:

(A) is greater

Explanation:

97 is the only prime number between 90 and 100, so N = 97 . The only two primes between 80 and 90 are 83 and 89, so M = 83 or M = 89 . Therefore, either of the following holds:

M + N = 83 + 97 = 180 > 175

M + N = 89 + 97 = 186 > 175

(A) must be the greater quantity regardless.

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Example Question #35 : How To Find Out If A Number Is Prime

and are prime numbers.

30 < M <40 < N < 50

Which is the greater quantity?

(A) M + N

(B) 80

Possible Answers:

(B) is greater

(A) and (B) are equal

(A) is greater

It is impossible to determine which is greater from the information given

Correct answer:

It is impossible to determine which is greater from the information given

Explanation:

We show that it cannot be determined whether M + N is greater than, less than, or equal to 80 by choosing two pairs of primes within the given ranges and adding.

Case 1:

$M = 31, N = 41 \Rightarrow M + N = 31+ 41 = 72 < 80$ .

Case 2:

$M = 37, N = 47 \Rightarrow M + N = 37+ 47 = 84 > 80$

In each case, 30 < M <40 < N < 50 , with and prime.

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Example Question #91 : Factors / Multiples

x+ y = 12 , and and are positive integers.

is a prime number; is not a prime number.

Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

It cannot be determined which of (a) and (b) is greater

Correct answer:

It cannot be determined which of (a) and (b) is greater

Explanation:

x+ y = 12 , and and are positive integers, so each of and is an integer from 1 to 11 inclusive.

is a prime number, meaning that it can be equal to 2, 3, 5, 7, or 11. Testing each case:

x=2

2+ y = 12

y = 10 , which is not prime.

x= 3

3+ y = 12

y = 9 , which is not prime.

x= 5

5+ y = 12

y = 7 , which is prime - we throw this case out.

x= 7

7+ y = 12

y = 5 , which is prime - we throw this case out.

x= 11

11+ y = 12

y = 1 , which is not prime.

In the first two cases, y > x ; in the last case, y < x . It cannot be determined which is the greater.

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Example Question #42 : How To Find Out If A Number Is Prime

x+ y = 10 ; and are positive integers.

is a prime number. is not.

Which is the greater quantity?

(a) 5

(b)

Possible Answers:

(a) and (b) are equal

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

x+ y = 10 , and and are positive integers.

Therefore, must be an integer from 1 to 9, as must .

Since is prime, it can be any of 2, 3, 5, or 7.

Therefore, one of the following must hold:

2+ y = 10

y = 8

3+ y = 10

y = 7

5+ y = 10

y = 5

7+ y = 10

y = 3

Only in the first case is not a prime number (8 has four factors - 1, 2, 4, 8), so x = 2 and y = 8 > 5 .

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Example Question #1 : Least Common Multiple

What is the least common multiple of 15 and 25?

Possible Answers:

125

175

150

Correct answer:

Explanation:

$\textrm{LCM }(15,25)$ is the lowest number that is a multiple of both 15 and 25, so we see which is the first number that appears in both lists of multiples.

The multiples of 15:

$\left \{ 15, 30, 45, 60, \underline{75}, 90, 105,...\right \}$

The multiples of 25:

$\left \{ 25, 50, \underline{75}, 100, 125, 150,...\right \}$

$\textrm{LCM }(15,25)= 75$

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Example Question #1 : Least Common Multiple

N= M + 1

Which is the greater quantity?

(a) LCM(M, 100)

(b) LCM(N, 100)

Possible Answers:

(a) is greater.

(a) and (b) are equal.

Not enough information is given to answer the question.

(b) is greater.

Correct answer:

Not enough information is given to answer the question.

Explanation:

We show that the given information is not enough by taking two cases:

M = 2, N = 3 and M = 3, N = 4

and divide into 100 , so LCM(2, 100) = 100 and LCM(4, 100) = 100 .

is prime and $100 = 2\times2\times5\times5$ , so

$LCM(3, 100) = 2\times2\times3\times5\times5 = 300$ .

Therefore, if M = 2, N = 3 , (b) is greater, and if M = 3, N = 4 , (a) is greater.

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Example Question #1 : How To Find The Least Common Multiple

Which is the greater quantity?

(a) $50 \times 60$

(b) $LCM (50,60 ) \times GCF (50,60 )$

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

The prime factorizations of 50 and 60 are:

$50 = 2 \times 5 \times 5$

$60 = 2\times2\times3\times5$

The greatest common factor of 50 and 60 is the product of the prime factors they share:

$GCF (50,60) = 2 \times 5 = 10$

The least common multiple of 50 and 60 is the product of all of the prime factors, with shared factors counted once:

$LCM (50,60 ) = 2\times2\times3\times5\times 5 = 300$

$GCF (50,60) \times LCM (50,60) = 10 \times 300 = 3,000 = 50 \times 60$ ,

(a) and (b) are equal.

Note: it is also a property of the integers that the product of the GCF and the LCM of two integers is equal to the product of the two integers themselves.

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Example Question #452 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Which is the greater quantity?

(a) $6 \times 8 \times 10$

(b) $GCF (6,8,10) \times LCM (6,8,10)$

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

Correct answer:

(a) is greater

Explanation:

(a) $6 \times 8 \times 10 = 480$

(b) To find GCF (6,8,10) , list their factors:

$6: 1, \underline{2}, 3, 6$

$8: 1,\underline{2}, 4, 8$

$10:1,\underline{2},5,10$

GCF (6,8,10) = 2

To find LCM (6,8,10) ,examine their prime factorizations:

6 = 2 $\times 3$

$8 = 2 \times 2 \times 2$

10: 2 $\times 5$

$LCM (6,8,10) = 2 \times 2 \times 2 \times 3 \times 5 = 120$

$GCF (6,8,10) \times LCM (6,8,10) = 2 \times 120 = 240$

(a) is greater.

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Example Question #2 : Least Common Multiple

Which of the following is the greater quantity?

(A) The least common multiple of 25 and 30

(B) 300

Possible Answers:

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(A) is greater

Correct answer:

(B) is greater

Explanation:

To find LCM(25,30) we can list some multiples of both numbers and discover the least number in both lists:

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

$30:30, 60, 90, 120, \underline{150}, 180...$

LCM(25,30) = 150 < 300 , so (B) is greater

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Example Question #3 : Least Common Multiple

Which of the following is the least common multiple of 25 and 40?

Possible Answers:

150

250

200

160

Correct answer:

200

Explanation:

List the first few multiples of both 25 and 40:

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

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

The least number in both lists of factors is 200.

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All ISEE Upper Level Quantitative Resources

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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 #33 : How To Find Out If A Number Is Prime

Example Question #35 : How To Find Out If A Number Is Prime

Example Question #91 : Factors / Multiples

Example Question #42 : How To Find Out If A Number Is Prime

Example Question #1 : Least Common Multiple

Example Question #1 : Least Common Multiple

Example Question #1 : How To Find The Least Common Multiple

Example Question #452 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #2 : Least Common Multiple

Example Question #3 : Least Common Multiple

All ISEE Upper Level Quantitative Resources

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