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In the order of operations, exponentiation comes first, then multiplication and division left to right, then addition and subtraction left to right. Here 4² = 16, 3 × 2 = 6, and 16 ÷ 8 = 2, so the expression becomes 12 − 6 + 2 = 8.
A number is divisible by 4 exactly when its last two digits form a number divisible by 4. The last two digits of 2,316 are 16, and 16 = 4 × 4, so 2,316 is divisible by 4.
List the integers greater than 20 and less than 40 that have no divisor other than 1 and themselves: 23, 29, 31, and 37. The other integers in that range are composite, for example 21 = 3 × 7, 25 = 5², and 33 = 3 × 11. So there are 4 such primes.
An absolute value equation splits into two cases: x − 3 = 7 gives x = 10, and x − 3 = −7 gives x = −4. The sum of the two solutions is 10 + (−4) = 6.
To take a percent of a number, convert the percent to a decimal and multiply: 0.15 × 240 = 36. Equivalently, 10% of 240 is 24 and 5% is 12, and 24 + 12 = 36.
The tenths digit is 9 and the hundredths digit is 8, which is 5 or more, so the tenths digit rounds up. Since 9 cannot be increased without carrying, 4.987 rounds to 5.0.
Rewrite each fraction with the common denominator 12: 9/12 + 10/12 − 4/12. Combining the numerators gives 15/12, which reduces to 5/4.
Write the decimal over its place value: 0.375 = 375/1000. Dividing numerator and denominator by their greatest common factor, 125, gives 3/8.
For any two positive integers, the product of the numbers equals the product of their greatest common factor and their least common multiple. So the product is 6 × 72 = 432, and the other number is 432 ÷ 24 = 18.
Write n = 7k + 4 for some integer k. Then 3n = 21k + 12 = 7(3k + 1) + 5, so dividing 3n by 7 leaves remainder 5. Equivalently, triple the original remainder, 3 × 4 = 12, and take 12 mod 7 = 5.
A 20% discount means the sale price is 80% of the original price. Solve 0.80P = 40 to get P = 40 ÷ 0.80 = 50, so the original price was $50.
Sharing 64 in the ratio 3 : 5 uses 3 + 5 = 8 equal parts, so each part is 64 ÷ 8 = 8. The numbers are 24 and 40, and their difference is 2 parts, or 16.
Adding two identical powers of 2 doubles the quantity, and doubling means adding 1 to the exponent. Since 2⁵ + 2⁵ = 2 × 2⁵ = 2⁶, and 2⁶ = 64.
A negative exponent takes the reciprocal of the base, so (2/3)⁻² = (3/2)². Squaring gives 9/4.
The square of an odd integer is odd, and adding 1 to an odd integer produces an even integer. Thus n² + 1 is always even when n is odd.
Multiply the coefficients and add the exponents: 3 × 2 = 6 and 10⁴ × 10⁻⁶ = 10⁻². The result is 6 × 10⁻², which equals 0.06.
Convert kilometers to meters by multiplying by 1,000 and hours to seconds by multiplying by 3,600. So the speed is 45 × 1000/3600 = 45 × 5/18 = 12.5 meters per second.
Set up the percent proportion p/100 = 40/320. Simplify 40/320 to 1/8, and 1/8 = 0.125, which is 12.5%.
The side length must divide both 84 and 60 exactly, so the largest possible side length is the greatest common factor. Since 84 = 2² × 3 × 7 and 60 = 2² × 3 × 5, the greatest common factor is 2² × 3 = 12.
A number is divisible by 9 exactly when the sum of its digits is divisible by 9. The digit sum is 3 + d + 7 = 10 + d, and the only single digit making 10 + d a multiple of 9 is d = 8, giving 18.
Apply the changes multiplicatively: 1.20 × 0.80 = 0.96. The final value is 96% of the original, which is a 4% decrease.
Factor 360 = 2³ × 3² × 5. The number of positive divisors equals the product of one more than each exponent: (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24.
Write 84 = 2² × 3 × 7. A perfect square must have even exponents on all primes, so the factor 3 and the factor 7 each need to be supplied once more. Thus the smallest n is 3 × 7 = 21, and 84 × 21 = 1764 = 42².
Units digits of powers of 7 repeat in the cycle 7, 9, 3, 1 with period 4. Since 45 divided by 4 leaves remainder 1, the units digit matches the first entry of the cycle, which is 7.
The sum of the first n positive odd integers equals n². With n = 50, the sum is 50² = 2500.
By the order of operations, multiplication and division are done before addition and subtraction. Thus 3 × 4 = 12 and 18 ÷ 6 = 3, so the expression becomes 20 − 12 + 3 = 11.
The absolute value of a number is its distance from zero. Here |−9| = 9 and |4 − 11| = |−7| = 7, so the difference is 9 − 7 = 2.
A number is divisible by 3 when its digit sum is divisible by 3, and by 4 when its last two digits form a multiple of 4. For 216, the digit sum is 9 and the last two digits are 16, so it is divisible by both.
The integers between 20 and 30 are 21 through 29. Checking divisibility shows that only 23 and 29 have no positive divisors other than 1 and themselves, so there are 2 such primes.
A fraction is closest to 1 when its difference from 1 is smallest. The differences are 1/12 ≈ 0.083, 1/10 = 0.1, 1/8 = 0.125, and 1/6 ≈ 0.167, so 11/12 is closest.
Rounding 0.49 to 0.5 and 81 to 80 gives 0.5 × 80 = 40 as an estimate. The exact product is about 39.69, so 40 is the closest choice.
On a number line, more negative numbers are farther to the left. Thus the correct increasing order is −7, then −3, then 0, then 2.
Percent increase is the amount of increase divided by the original amount, then converted to a percent. The increase is $12, and 12 ÷ 80 = 0.15, which is 15%.
In a ratio of 4 to 7, each part has the same size. Since 84 ÷ 7 = 12 marbles per part, the number of red marbles is 4 × 12 = 48.
The next simultaneous flash occurs after the least common multiple of 6 and 9. Since 6 = 2 × 3 and 9 = 3², the least common multiple is 2 × 3² = 18.
Write n = 11k + 5. Then n + 8 = 11k + 13 = 11(k + 1) + 2, so the remainder upon division by 11 is 2.
When multiplying powers with the same base, add exponents; when dividing, subtract exponents. Thus 2⁶ × 2⁴ = 2¹⁰ and 2¹⁰ ÷ 2⁷ = 2³ = 8.
Factor 72 as 36 × 2, and use the rule √(ab) = √a × √b. Then √72 = √36 × √2 = 6√2.
To take 5/9 of 36, first divide 36 by 9 to get 4 students per ninth, then multiply by 5. Thus 4 × 5 = 20 students walk.
Multiply 6 by 4 to get 24, then place the decimal point so that the result has three decimal places, since 0.6 has one and 0.04 has two. Thus 0.6 × 0.04 = 0.024.
Divide the coefficients and subtract the exponents: 6 ÷ 3 = 2 and 10⁷ ÷ 10² = 10⁵. The result is 2 × 10⁵.
For consecutive integers, the mean equals the average of the smallest and largest values. Thus the mean is (1 + 49) ÷ 2 = 25.
To convert inches to yards, divide the number of inches by 36. Since 252 ÷ 36 = 7, there are 7 yards.
The ratio 3 to 2 means each part is 15 ÷ 3 = 5 cups. Therefore the sugar required is 2 × 5 = 10 cups.
A number with prime factorization p^a q^b has (a+1)(b+1) divisors. The smallest number with 6 divisors has form p⁵ or p²q; checking the smallest primes gives 12 = 2² × 3, which has 3 × 2 = 6 divisors.
The powers of 3 modulo 5 repeat in the cycle 3, 4, 2, 1 with period 4. Since 100 is a multiple of 4, 3¹⁰⁰ leaves the same remainder as 3⁴, which is 1.
Among any three consecutive integers, at least one is even and one is a multiple of 3. Therefore the product contains factors of both 2 and 3, so it is always divisible by 6.
By inclusion–exclusion, the number who speak at least one language is 35 + 25 − 12 = 48. Subtracting from the total gives 60 − 48 = 12 people who speak neither.
A number is relatively prime to 10 exactly when it is divisible by neither 2 nor 5. In each block of 10 consecutive integers there are 4 such numbers, and there are 10 blocks from 1 to 99, giving 10 × 4 = 40.
For a positive constant k, the equation |A| = k splits into two cases: A = k and A = −k. Here 2x − 5 = 11 gives x = 8, and 2x − 5 = −11 gives x = −3, so the two possible values sum to 8 + (−3) = 5.