Correct: A. 4725

4725 is divisible by 9 because the sum of its digits is 4 + 7 + 2 + 5 = 18, and 18 is divisible by 9. The divisibility test for 9 allows a number to be checked by adding its digits rather than performing the full division. Since 18 is a multiple of 9, 4725 is also a multiple of 9.

Correct: A. 11

The remainder when 157 is divided by 12 is 1, because 12 × 13 = 156 and 157 is one greater than 156. In division with remainder, the remainder must be smaller than the divisor, so 1 is the valid remainder. This can also be written as 157 = 12 × 13 + 1.

Correct: C. 61

61 is prime because its only positive factors are 1 and 61. To verify this efficiently, it is enough to test prime divisors up to the square root of 61, which is less than 8, so 2, 3, 5, and 7 can be checked. None divides 61 evenly, confirming that 61 is prime.

Correct: C. sqrt(2)

The square root of 2 is irrational because it cannot be expressed as a ratio of two integers. In contrast, 0.75 and 7/8 are rational numbers, and every integer such as -3 is also rational because it can be written over 1. Irrational numbers have decimal expansions that neither terminate nor repeat periodically.

Correct: C. 3

The smallest number that must be added to 875 is 3 because the digit sum of 875 is 20, which leaves a remainder of 2 when divided by 9. Adding 3 changes the number to 878, whose digit sum is 23, so that does not work, indicating the direct digit-sum approach needs correction. In fact, 875 divided by 9 leaves a remainder of 2, so adding 7 gives 882, which is divisible by 9.

Correct: C. 72

72 is divisible by both 4 and 6 because 72 ÷ 4 = 18 and 72 ÷ 6 = 12. A number divisible by both must be a common multiple of the two divisors. The least common multiple of 4 and 6 is 12, and 72 is a multiple of 12.

Correct: C. 25

The first five positive odd integers are 1, 3, 5, 7, and 9, and their sum is 25. Adding them gives 1 + 3 + 5 + 7 + 9 = 25. The sum of the first n odd positive integers has the useful pattern that it equals n².

Correct: B. 81

81 is a perfect square because 81 = 9². A perfect square is an integer that can be expressed as the square of another integer, and 9 × 9 produces 81. The neighboring numbers 72, 90, and 98 do not have integer square roots.

Correct: C. 16

There are 16 integers from 25 through 40 inclusive. The count is found by subtracting the endpoints and adding 1, giving 40 ? 25 + 1 = 16. The additional 1 is necessary because both endpoint values are included.

Correct: C. 2

The value is 2 because every nonzero number raised to the zero power equals 1, so 5? + 3? = 1 + 1 = 2. The zero-exponent property applies to any nonzero base. It is important that the bases here are nonzero because 0? is not defined under the usual elementary exponent rules.

Correct: C. 0.875

The fraction 7/8 is equal to 0.875 because dividing 7 by 8 gives 0.875. Since 8 is a power of 2, the decimal terminates after a finite number of places. Converting between fractions and decimals is useful for comparing quantities represented in different numerical forms.

Correct: C. 6

A number divisible by both 2 and 3 must also be divisible by 6 because 2 and 3 are relatively prime and their product is 6. For example, 18 is divisible by both 2 and 3 and also by 6. This principle follows from the relationship between common multiples and the least common multiple.

Correct: B. 105

The smallest three-digit number divisible by 7 is 105 because 7 × 15 = 105. The preceding multiple is 7 × 14 = 98, which is only two digits, so 105 is the first three-digit multiple of 7. Multiples are obtained by multiplying a number by successive positive integers.

Correct: A. 10^7

The value is 10? because powers with the same base are multiplied by adding their exponents, so 10³ × 10? = 10^(3+4) = 10?. This exponent rule is called the product property of exponents. It avoids expanding the powers into large numbers before multiplying them.

Correct: B. 53

The smallest prime number greater than 50 is 53 because 53 has no positive divisors other than 1 and itself. The numbers 51, 55, and 57 are composite because they are divisible by 3, 5, and 3 respectively. Checking possible prime divisors up to the square root of a number is enough to establish primality.

Correct: D. 39

39 is composite because it has factors other than 1 and itself, specifically 3 and 13. The other choices, 29, 31, and 37, are prime numbers because they have no positive divisors besides 1 and the number itself. A composite number can always be expressed as a product of two smaller positive integers.

Correct: B. -4

?4 is an integer but not a whole number because integers include negative numbers while whole numbers consist of 0 and the positive counting numbers. The set of integers includes ..., ?3, ?2, ?1, 0, 1, 2, 3, ..., whereas whole numbers do not include negative values. Therefore, ?4 belongs to the integers but not to the whole numbers.

Correct: A. 17

The value is 23 because 2^5 = 32 and 3^2 = 9, giving 32 ? 9 = 23. Exponents are evaluated before subtraction according to the standard order of operations. Applying the exponent rule correctly prevents the bases and exponents from being confused with ordinary multiplication.

Correct: B. 15

The value of x is 15 because dividing both sides of 3x = 45 by 3 gives x = 15. This uses the inverse relationship between multiplication and division. The resulting value can be checked by substituting it back into the original equation, giving 3 × 15 = 45.

Correct: B. 24

24 is a multiple of both 8 and 12 because 24 = 8 × 3 and 24 = 12 × 2. A common multiple is a number that can be divided exactly by each of the given numbers. Although 48 is also a common multiple, 24 is the first such option listed that satisfies both conditions.

Correct: A. 3

The digit in the thousands place is 3 because 583,721 can be separated by place value as 500,000 + 80,000 + 3,000 + 700 + 20 + 1. The thousands place is the fourth position from the right in a six-digit whole number. Place value determines the numerical value contributed by each digit based on its position.

Correct: C. 17/28

17/28 is already in lowest terms because 17 and 28 have no common factor greater than 1. The numerator 17 is prime, and it does not divide the denominator 28, so the fraction cannot be reduced further. A fraction is in lowest terms when its numerator and denominator have a greatest common factor of 1.

Correct: B. 4-Jan

Twenty-five percent is equal to 1/4 because 25% means 25 out of 100, giving 25/100, which simplifies by dividing both numbers by 25. The resulting fraction is 1/4. Percentages, fractions, and decimals are different representations of the same numerical quantity.

Correct: C. 975

975 is divisible by 5 because its last digit is 5. A whole number is divisible by 5 exactly when its final digit is either 0 or 5. This divisibility rule allows a quick check without performing the complete division.

Correct: C. 55

The sum of the first ten positive integers is 55. Adding 1 through 10 gives 55, and the same result follows from the formula n(n + 1)/2 with n = 10. This formula provides an efficient way to add consecutive positive integers without listing every term.

Correct: B. Its last digit is 0

A number divisible by 10 must have a last digit of 0. This follows because every multiple of 10 has the form 10k, which shifts the digits of k one place to the left and adds a zero. The condition is necessary for divisibility by 10 in the decimal number system.

Correct: B. 24

The value is 24 because 2^3 = 8 and 4^2 = 16, and 8 + 16 = 24. The exponentiation must be performed before the addition according to the standard order of operations. Evaluating the powers separately makes the calculation straightforward.

Correct: C. 7

The remainder when 999 is divided by 8 is 7 because 8 × 124 = 992 and 999 ? 992 = 7. The remainder must be smaller than the divisor, so 7 is valid while 8 cannot be a remainder when dividing by 8. This calculation can also be checked by noting that 999 is one less than 1000, which is divisible by 8.

Correct: B. 64

64 is a perfect cube because 64 = 4^3. A perfect cube is an integer that can be written as the cube of another integer, and 4 × 4 × 4 equals 64. The other listed numbers are not cubes of integers.

Correct: C. 9

The value is 9 because dividing powers with the same nonzero base means subtracting their exponents, so 3^4 ÷ 3^2 = 3^(4?2) = 3² = 9. This exponent rule is called the quotient property of exponents. It simplifies calculations without requiring the full powers to be computed first.

Correct: B. 234

234 is divisible by 6 because it is divisible by both 2 and 3. Its last digit is 4, making it even, and its digit sum is 2 + 3 + 4 = 9, which is divisible by 3. A number is divisible by 6 exactly when it is divisible by both 2 and 3.

Correct: C. 12

The remainder is 2 because 15 × 23 = 345 and 347 ? 345 = 2. Therefore, 347 can be expressed as 15 × 23 + 2. The remainder is always smaller than the divisor, so 2 is a valid remainder for division by 15.

Correct: A. 121

121 is a perfect square because 121 = 11^2. A perfect square is an integer that can be expressed as the product of an integer with itself. The other listed values do not have integer square roots.

Correct: A. 125

125 is a perfect cube because 125 = 5^3. A perfect cube is an integer that can be written as the cube of another integer, and 5 × 5 × 5 equals 125. The other options are not cubes of integers.

Correct: D. 15/30

15/30 is equivalent to 3/5 because dividing both numerator and denominator by 5 gives 3/6, so this option is not correct; therefore this question is invalid as written.

Correct: A. 104

The smallest three-digit multiple of 13 is 104 because 13 × 8 = 104. The preceding multiple is 13 × 7 = 91, which has only two digits, so 104 is the first three-digit multiple. Multiples are obtained by multiplying a number by consecutive whole numbers.

Correct: A. 432

432 is divisible by 9 because its digits sum to 4 + 3 + 2 = 9, and 9 is divisible by 9. The divisibility test for 9 allows the digit sum to be used instead of performing the full division. The other listed numbers have digit sums that are not multiples of 9.

Correct: C. 25

The value is 25 because dividing powers with the same nonzero base means subtracting their exponents: 5^3 ÷ 5^1 = 5^(3?1) = 5^2 = 25. The quotient property of exponents simplifies expressions without requiring the full power to be calculated first. The divisor 5 can be written as 5^1 to make the exponent rule explicit.

Correct: B. ?2

?2 is an integer because integers include negative whole values, zero, and positive whole values. The fraction 3/4 and decimal 1.5 are not integers, while ?2 is irrational and therefore not an integer. Integers are commonly represented by the set {..., ?2, ?1, 0, 1, 2, ...}.

Correct: B. 42

The first six positive even integers are 2, 4, 6, 8, 10, and 12, whose sum is 42. Adding these values directly gives 42, and the result also follows from the formula n(n + 1) for the sum of the first n positive even integers. This pattern is useful when working with consecutive even numbers.

Correct: B. 101

The value is 110 because 10^2 = 100 and 10^1 = 10, giving 100 + 10 = 110. Powers of ten are particularly straightforward because each exponent indicates the number of factors of 10 being multiplied. This makes place-value calculations and scientific notation easier to interpret.

Correct: B. 23

23 has exactly two positive factors, 1 and 23, so it is prime. A prime number is defined as a positive integer greater than 1 with exactly two positive factors. The other choices are composite because each has additional factors beyond 1 and itself.