Test: Number Properties - GMAT MCQ

If 2 and 17 are factors of a positive integer n, then any multiple of their product must also be a factor of n. The product of 2 and 17 is 34.

Now let's examine each option:

I. 34
Since 34 is the product of 2 and 17, it must also be a factor of n.

II. 68
68 is not a multiple of 34. Therefore, it does not necessarily have to be a factor of n.

III. 136
136 is twice the value of 68 and is not a multiple of 34. Therefore, it does not necessarily have to be a factor of n.

Based on this analysis, the only option that must divide into n is I: 34.

Therefore, the correct answer is A: I only.

To find the number of odd integers between 10/3 and 62/3, we can examine the range and count the odd integers within that range.

The given range is from 10/3 to 62/3.

To determine the number of odd integers, we need to find the number of integers between the smallest and largest integer values within the range.

10/3 is equivalent to 3 and 1/3, while 62/3 is equivalent to 20 and 2/3.

The smallest integer value within the range is 4, obtained by rounding up 3 and 1/3. The largest integer value within the range is 20, obtained by rounding down 20 and 2/3.

We need to count the number of odd integers between 4 and 20 (inclusive).

The odd integers within this range are 5, 7, 9, 11, 13, 15, 17, 19.

Therefore, there are a total of 8 odd integers between 10/3 and 62/3.

The correct answer is E: Eight.

If x is the sum of six consecutive integers, we can represent these integers as x - 2, x - 1, x, x + 1, x + 2, and x + 3 (from the smallest to the largest integer).

Now let's examine each option:

I. 3 The consecutive integers have a common difference of 1, so the sum of six consecutive integers will be a multiple of 3. Therefore, x is divisible by 3.

II. 4 The sum of six consecutive integers does not necessarily guarantee divisibility by 4. For example, the consecutive integers could be -3, -2, -1, 0, 1, and 2, which do not add up to a multiple of 4. Therefore, x is not necessarily divisible by 4.

III. 6 The sum of six consecutive integers does not necessarily guarantee divisibility by 6. For example, the consecutive integers could be -2, -1, 0, 1, 2, and 3, which do not add up to a multiple of 6. Therefore, x is not necessarily divisible by 6.

Based on this analysis, the only option that x must be divisible by is I: 3.

Therefore, the correct answer is A: I only.

To find the greatest prime factor of the sum of the numbers from 1 to 36, we can calculate the sum first and then determine its prime factors.

The sum of the numbers from 1 to 36 can be calculated using the formula for the sum of an arithmetic series:

Sum = (n/2)(first term + last term)

In this case, the first term is 1, the last term is 36, and there are 36 terms.

Sum = (36/2)(1 + 36) = 18(37) = 666

Now, let's determine the prime factors of 666.

666 can be factored as 2 * 3 * 3 * 37.

The greatest prime factor is 37.

Therefore, the correct answer is E: 37.

To find the number of prime numbers between 50 and 70, we can examine each number within this range and determine if it is a prime number.

The numbers between 50 and 70 (inclusive) are: 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70.

Now, let's determine which of these numbers are prime:

51 is not prime as it is divisible by 3 (3 * 17).

52 is not prime as it is divisible by 2 (2 * 26).

53 is a prime number.

54 is not prime as it is divisible by 2 (2 * 27).

55 is not prime as it is divisible by 5 (5 * 11).

56 is not prime as it is divisible by 2 (2 * 28).

57 is not prime as it is divisible by 3 (3 * 19).

58 is not prime as it is divisible by 2 (2 * 29).

59 is a prime number.

60 is not prime as it is divisible by 2 (2 * 30) and 3 (3 * 20).

61 is a prime number.

62 is not prime as it is divisible by 2 (2 * 31).

63 is not prime as it is divisible by 3 (3 * 21).

64 is not prime as it is divisible by 2 (2 * 32).

65 is not prime as it is divisible by 5 (5 * 13).

66 is not prime as it is divisible by 2 (2 * 33) and 3 (3 * 22).

67 is a prime number.

68 is not prime as it is divisible by 2 (2 * 34).

69 is not prime as it is divisible by 3 (3 * 23).

70 is not prime as it is divisible by 2 (2 * 35) and 5 (5 * 14).

From the analysis, we can see that there are four prime numbers between 50 and 70: 53, 59, 61, and 67.

Therefore, the correct answer is C: 4.

To determine the value of x that may not be equal to the given equation when rounded to the nearest integer, we can substitute each value into the equation y = 5x and round it to the nearest integer. Let's examine each option:

I. x = 2.9
Substituting x = 2.9 into y = 5x:
y = 5 * 2.9 = 14.5
Rounding 14.5 to the nearest integer gives 15, which matches the given result. Therefore, x = 2.9 is a valid solution.

II. x = 2.95
Substituting x = 2.95 into y = 5x:
y = 5 * 2.95 = 14.75
Rounding 14.75 to the nearest integer gives 15, which matches the given result. Therefore, x = 2.95 is a valid solution.

III. x = 3.1
Substituting x = 3.1 into y = 5x:
y = 5 * 3.1 = 15.5
Rounding 15.5 to the nearest integer gives 16, which does not match the given result of 15. Therefore, x = 3.1 is NOT a valid solution.

Based on this analysis, the value of x that may NOT be equal to the given equation when rounded to the nearest integer is III. Therefore, the correct answer is B: III only.

To find the number of different positive prime factors of N, we need to consider the given condition: 14N/60 is an integer.

We can simplify the expression 14N/60 by dividing both the numerator and denominator by their greatest common divisor (GCD). The GCD of 14 and 60 is 2.

(14N/2) / (60/2) = 7N/30

For 7N/30 to be an integer, N must be a multiple of 30 because the denominator 30 contains the prime factors 2, 3, and 5.

Now let's examine the prime factorization of 30:

30 = 2 * 3 * 5

The prime factorization of 30 consists of three distinct prime factors: 2, 3, and 5.

Therefore, the correct answer is B: 3. N has three different positive prime factors: 2, 3, and 5.

If n + 12 is a positive odd integer, it means that n is an odd integer. Let's consider the next four integers after n.

The four consecutive integers after n can be represented as n + 1, n + 2, n + 3, and n + 4.

Since n is odd, all four consecutive integers will be odd as well, as adding an even number to an odd number results in an odd number.

The sum of two odd numbers is always an even number. Therefore, the sum of n + 1, n + 2, n + 3, and n + 4 will be an even number.

Hence, the correct answer is E: It is a multiple of 2.

To determine which integer does not have a divisor greater than 1 that is the square of an integer, we need to examine each option.

A: The number 75 can be factored as 3 * 5 * 5. It has a divisor greater than 1 that is the square of an integer, which is 5^2 = 25.

B: The number 42 can be factored as 2 * 3 * 7. Let's check if it has a divisor greater than 1 that is the square of an integer. We see that none of the prime factors, 2, 3, or 7, are squared. Therefore, 42 does not have a divisor greater than 1 that is the square of an integer.

C: The number 32 can be factored as 2 * 2 * 2 * 2 * 2. It has a divisor greater than 1 that is the square of an integer, which is 2^2 = 4.

D: The number 25 can be factored as 5 * 5. It has a divisor greater than 1 that is the square of an integer, which is 5^2 = 25.

E: The number 12 can be factored as 2 * 2 * 3. It has a divisor greater than 1 that is the square of an integer, which is 2^2 = 4.

Therefore, the only option that does not have a divisor greater than 1 that is the square of an integer is B: 42.

To determine the number of different positive numbers that divide the product XYZ, we need to consider the prime factorization of XYZ.

Since X, Y, and Z are three different prime numbers, their product XYZ is the product of three distinct primes. Let's represent X, Y, and Z as follows:

X = p₁
Y = p₂
Z = p₃

Here, p₁, p₂, and p₃ are three distinct prime numbers.

The prime factorization of XYZ is given by:

XYZ = (p₁)(p₂)(p₃)

Since X, Y, and Z are prime numbers, they have no factors other than 1 and themselves. Therefore, the prime factorization of XYZ contains only the three primes p₁, p₂, and p₃, and no other prime factors.

To find the number of different positive numbers that divide XYZ, we count the number of factors in the prime factorization. For a number with prime factorization p₁^a * p₂^b * p₃^c, where a, b, and c are positive integers, the number of factors is given by (a + 1)(b + 1)(c + 1).

In this case, the prime factorization of XYZ is (p₁)(p₂)(p₃), so the number of factors is (1 + 1)(1 + 1)(1 + 1) = 2 * 2 * 2 = 8.

Therefore, the correct answer is C: 8.