ISEE Upper Level Quantitative Reasoning - Numbers and Operations

Which is the greater quantity?

(a) The number of factors of 169

(b) The number of factors of 121

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

Each number has only three factors. 121 has 1, 11, and 121 as factors; 169 has 1, 13, and 169 as factors. The answer is that the quantities are equal.

Define an operation $\bigstar$ as follows:

For all positive integers and

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

Evaluate $100 \bigstar 80$ .

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:

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

Which is the greater quantity?

(a) The number of prime numbers between 1 and 20 inclusive

(b) The number of composite numbers between 21 and 30 inclusive

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

(a) The prime numbers between 1 and 20 inclusive are 2, 3, 5, 7, 11, 13, 17, 19 - eight total.

(b) The prime numbers between 21 and 30 inclusive are 23 and 29 - two prime numbers out of ten integers. This leaves eight composite numbers.

(a) and (b) are therefore equal.

Which is the greater quantity?

(a) The number of prime numbers between 70 and 110

(b) The number of prime numbers between 80 and 120

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

The primes between 80 and 110 are included in both sets, so all we need to do is to compare the number of primes between 70 and 80 and the number of primes between 110 and 120.

(a) The primes between 70 and 80 are 71, 73, and 79 - three primes

(b) The only prime between 110 and 120 is 113.

(a) is the greater quantity

Which is the greater quantity?

(a) The number of prime numbers between 1 and 20 inclusive

(b) The number of composite numbers between 21 and 30 inclusive

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

(a) The prime numbers between 1 and 20 inclusive are 2, 3, 5, 7, 11, 13, 17, 19 - eight total.

(b) The prime numbers between 21 and 30 inclusive are 23 and 29 - two prime numbers out of ten integers. This leaves eight composite numbers.

(a) and (b) are therefore equal.

Define an operation $\bigstar$ as follows:

For all positive integers and

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

Evaluate $100 \bigstar 80$ .

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:

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

$\frac{ \left ( 30 -2 \cdot 15\right )^{0}}{\left ( 30 -15\right )^{0}}$

The expression is undefined.

Explanation

$\frac{ \left ( 30 -2 \cdot 15\right )^{0}}{\left ( 30 -15\right )^{0}} = \frac{ \left ( 30 -30\right )^{0}}{15^{0}}=\frac{ 0^{0}}{15^{0}}$

The numerator is undefined, since 0 raised to the power of 0 is an undefined quantity. Therefore, the entire expression is undefined.

Which is the greater quantity?

(a) The number of prime numbers between 1 and 20 inclusive

(b) The number of composite numbers between 21 and 30 inclusive

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

(a) The prime numbers between 1 and 20 inclusive are 2, 3, 5, 7, 11, 13, 17, 19 - eight total.

(b) The prime numbers between 21 and 30 inclusive are 23 and 29 - two prime numbers out of ten integers. This leaves eight composite numbers.

(a) and (b) are therefore equal.

Which is the greater quantity?

(a) The number of factors of 169

(b) The number of factors of 121

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

Each number has only three factors. 121 has 1, 11, and 121 as factors; 169 has 1, 13, and 169 as factors. The answer is that the quantities are equal.

Which is the greater quantity?

(a) The number of prime numbers between 70 and 110

(b) The number of prime numbers between 80 and 120

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

The primes between 80 and 110 are included in both sets, so all we need to do is to compare the number of primes between 70 and 80 and the number of primes between 110 and 120.

(a) The primes between 70 and 80 are 71, 73, and 79 - three primes

(b) The only prime between 110 and 120 is 113.

(a) is the greater quantity

Numbers and Operations

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