GMAT Math : Understanding the properties of integers

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Example Question #1 : Understanding The Properties Of Integers

What is the sum of the prime numbers that are greater than 50 but less than 60?

Possible Answers:

$\dpi{100} \small 112$ $\displaystyle \dpi{100} \small 112$

$\dpi{100} \small 110$ $\displaystyle \dpi{100} \small 110$

$\dpi{100} \small 173$ $\displaystyle \dpi{100} \small 173$

$\dpi{100} \small 51$ $\displaystyle \dpi{100} \small 51$

$\dpi{100} \small 179$ $\displaystyle \dpi{100} \small 179$

Correct answer:

$\dpi{100} \small 112$ $\displaystyle \dpi{100} \small 112$

Explanation:

A prime number is only divisible by the number 1 and itself. Of the integers between 50 and 60, all of the even integers are also divisible by the number 2 so they are not prime numbers. The integers 51 and 57 are divisible by 3. The integer 55 is divisible by 5. The only integers that are prime are 53 and 59. The sum of these two integers is 112.

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Example Question #1 : Understanding The Properties Of Integers

What is the product of the four smallest prime numbers?

Possible Answers:

210

945

1155

Correct answer:

210

Explanation:

We must remember that 0 and 1 are NOT prime numbers, but 2 is.

The four smallest prime numbers are 2, 3, 5, and 7. Then 2 * 3 * 5 * 7 = 210.

Note: There are NO negative prime numbers, so we don't have to look for tiny, negative numbers here.

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Example Question #2 : Properties Of Integers

If a and b are even integers, what must be odd?

Possible Answers:

$\dpi{100} \small a+b-2$ $\displaystyle \dpi{100} \small a+b-2$

$\dpi{100} \small a\times b$ $\displaystyle \dpi{100} \small a\times b$

$\dpi{100} \small a+b$ $\displaystyle \dpi{100} \small a+b$

$\dpi{100} \small a-b$ $\displaystyle \dpi{100} \small a-b$

$\dpi{100} \small a+b-1$ $\displaystyle \dpi{100} \small a+b-1$

Correct answer:

$\dpi{100} \small a+b-1$ $\displaystyle \dpi{100} \small a+b-1$

Explanation:

The sum (or difference) or 2 even integers is even. Similarly, the product (or quotient) of 2 even integers is also even; therefore the answer must be $\dpi{100} \small a+b-1$ $\displaystyle \dpi{100} \small a+b-1$, which can be easily checked by plugging in any two even numbers.

For example, if $\dpi{100} \small a=2\ and\ b=4,\ a+b-1=2+4-1=5$ $\displaystyle \dpi{100} \small a=2\ and\ b=4,\ a+b-1=2+4-1=5$, which is odd.

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Example Question #1 : Properties Of Integers

Three consecutive numbers add up to 36. What is the smallest number?

Possible Answers:

$\displaystyle 9$

$\displaystyle 11$

$\displaystyle 12$

$\displaystyle 13$

$\displaystyle 10$

Correct answer:

$\displaystyle 11$

Explanation:

The sum of 3 consecutive numbers would be

$\displaystyle n + (n +1) + (n+2)$

which simplifies into

3n +3 $\displaystyle 3n +3$

Set that equation equal to 36 and solve.

$3n + 3 = 36 \rightarrow 3n = 33 \rightarrow n = 11$ $\displaystyle 3n + 3 = 36 \rightarrow 3n = 33 \rightarrow n = 11$

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Example Question #1 : Understanding The Properties Of Integers

Becky has to choose from 4 pairs of pants, 6 shirts, and 2 pairs of shoes for an interview. If an outfit consists of 1 pair of pants, 1 pair of shoes, and 1 shirt, how many options does she have?

Possible Answers:

$\displaystyle 48$

$\displaystyle 24$

$\displaystyle 60$

120 $\displaystyle 120$

Correct answer:

$\displaystyle 48$

Explanation:

To find total number, multiply the number of each item.

$\displaystyle (6)(4)(2)=48$

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Example Question #3 : Properties Of Integers

What is the greatest prime factor of $\displaystyle (3^7)(5^8)?$

Possible Answers:

$\displaystyle 9$

$\displaystyle 5$

$\displaystyle 7$

$\displaystyle 3$

Correct answer:

$\displaystyle 5$

Explanation:

The only prime factors are 3 and 5, therefore, 5 will be the greatest prime factor.

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Example Question #1 : Understanding The Properties Of Integers

Solve:

$(8\times 10^6)+(5\times 10^4)+(3\times 10^3)+(4\times 10^2)+(6\times 10)+(7\times 10^0)=$ $\displaystyle (8\times 10^6)+(5\times 10^4)+(3\times 10^3)+(4\times 10^2)+(6\times 10)+(7\times 10^0)=$

Possible Answers:

$\displaystyle 8,053,467$

853,467 $\displaystyle 853,467$

$\displaystyle 80,053,467$

$\displaystyle 800,053,467$

Correct answer:

$\displaystyle 8,053,467$

Explanation:

$8\times 10^6=8,000,000$ $\displaystyle 8\times 10^6=8,000,000$

$5\times 10^4=50,000$ $\displaystyle 5\times 10^4=50,000$

$3\times 10^3=3,000$ $\displaystyle 3\times 10^3=3,000$

$4\times 10^2=400$ $\displaystyle 4\times 10^2=400$

$6\times 10=60$ $\displaystyle 6\times 10=60$

$7\times 10^0=7$ $\displaystyle 7\times 10^0=7$

The sum is 8,053,467

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Example Question #2 : Understanding The Properties Of Integers

If a positive integer $\displaystyle N$ is divided by another positive integer $\displaystyle M$, then the quotient is 6, and there is no remainder.

Which of these choices is a possible value of M + N $\displaystyle M + N$?

Possible Answers:

$\displaystyle 91$

784 $\displaystyle 784$

392 $\displaystyle 392$

Each of the other choices is a possible value of M + N $\displaystyle M + N$.

$\displaystyle 77$

Correct answer:

Each of the other choices is a possible value of M + N $\displaystyle M + N$.

Explanation:

The conditions of the problem can be rewritten as $N \div M = 6$ $\displaystyle N \div M = 6$, or N =6M $\displaystyle N =6M$.

$\displaystyle M + N = M + 6M = 7M$, meaning that the sum of the two numbers is a multiple of 7. Each of the given choices is a multiple of 7, so any of them can be M + N $\displaystyle M + N$.

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Example Question #1 : Understanding The Properties Of Integers

What is the first digit in the base-six representation of the number 936?

Possible Answers:

$\displaystyle 2$

$\displaystyle 4$

$\displaystyle 3$

$\displaystyle 5$

$\displaystyle 1$

Correct answer:

$\displaystyle 4$

Explanation:

$\displaystyle 216 < 936 < 1296$, or, equivalently, $6^{3}< 936 < 6^{4}$ $\displaystyle 6^{3}< 936 < 6^{4}$. This makes 936 a four-digit number when written in base six. The first digit is equal to the number of times 216 divides into 936.

$936 \div 216 = 4 \textrm{ R } 72$ $\displaystyle 936 \div 216 = 4 \textrm{ R } 72$,

4 is the first digit of this number.

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Example Question #1 : Understanding The Properties Of Integers

What is the last digit in the base-eight representation of the number 735?

Possible Answers:

$\displaystyle 1$

$\displaystyle 0$

$\displaystyle 7$

$\displaystyle 5$

$\displaystyle 3$

Correct answer:

$\displaystyle 7$

Explanation:

Divide 735 by 8. The remainder is the last digit of the base-eight representation.

$735 \div 8 = 91 \textrm{ R } 7$ $\displaystyle 735 \div 8 = 91 \textrm{ R } 7$

The correct choice is 7.

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GMAT Math : Understanding the properties of integers

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Example Questions

Example Question #1 : Understanding The Properties Of Integers

Example Question #1 : Understanding The Properties Of Integers

Example Question #2 : Properties Of Integers

Example Question #1 : Properties Of Integers

Example Question #1 : Understanding The Properties Of Integers

Example Question #3 : Properties Of Integers

Example Question #1 : Understanding The Properties Of Integers

Example Question #2 : Understanding The Properties Of Integers

Example Question #1 : Understanding The Properties Of Integers

Example Question #1 : Understanding The Properties Of Integers

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