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In a base-ten numeral each position has a place value, and moving left multiplies the value by ten. The 7 sits in the hundreds place, so it represents 7 × 100 = 700.
A prime has exactly two positive divisors, 1 and itself. The number 53 is not divisible by 2, 3, 5 or 7, and since 7² exceeds 53 no larger factor needs testing, so 53 is prime.
Rewrite both fractions with the common denominator 12: 3/4 = 9/12 and 5/6 = 10/12. Adding numerators gives 19/12, which is already in lowest terms because 19 is prime.
Factor both numbers: 36 = 2² × 3² and 60 = 2² × 3 × 5. The GCD takes the lowest power of each shared prime, giving 2² × 3 = 12.
A 25 percent discount removes one quarter of the original price: 80 × 0.25 = 20 dollars. Subtracting that from 80 leaves a sale price of 60 dollars.
A number is divisible by 9 exactly when the sum of its digits is a multiple of 9. For 4,563 the digit sum is 4+5+6+3 = 18, which is divisible by 9, so 4,563 is divisible by 9.
When multiplying powers with the same base, add the exponents to get 2⁷. Dividing by 2⁵ subtracts exponents, giving 2⁷⁻⁵ = 2², which equals 4.
Set up the proportion 5/2 = 35/x, where x is the sugar mass. Cross-multiplying gives 5x = 70, so x = 14 grams.
Write 12 = 2² × 3 and 18 = 2 × 3². The LCM uses the highest power of each prime, giving 2² × 3² = 36, the smallest number divisible by both.
Work inside the parentheses first: 4 − 6 = −2. Then multiply and divide left to right: 3 × (−2) = −6 and −6 ÷ 2 = −3. Finally 12 − (−3) = 15.
Write the number as 7k + 3. Then three times it is 21k + 9 = 7(3k + 1) + 2, so the remainder is 2. Equivalently, 3 × 3 = 9 leaves remainder 2 modulo 7.
Factor 72 = 2³ × 3². A divisor has the form 2^a × 3^b with 0 ≤ a ≤ 3 and 0 ≤ b ≤ 2, giving 4 × 3 = 12 choices.
Let x = 0.4444... Then 10x = 4.4444..., and subtracting gives 9x = 4, so x = 4/9. This fraction is already in lowest terms.
After a 20 percent increase the price is 1.2 times the original, and a 20 percent decrease multiplies by 0.8. The product 1.2 × 0.8 = 0.96 means the final price is 96 percent of the original, a 4 percent decrease.
The units digits of powers of 3 cycle 3, 9, 7, 1 with period 4. Since 2025 = 4 × 506 + 1, the remainder is 1, so the units digit matches the first entry of the cycle, which is 3.
Scientific notation writes a number as a × 10ⁿ with 1 ≤ a < 10. Moving the decimal point four places right turns 0.00036 into 3.6, so the exponent is −4.
For two positive integers, the product of the GCD and LCM equals the product of the numbers. So 6 × 90 = 540 = 18 × x, giving x = 30, and indeed gcd(18, 30) = 6 and lcm(18, 30) = 90.
Use inclusion–exclusion: 75 multiples of 4, 50 multiples of 6, and 25 multiples of 12 counted twice. Thus 75 + 50 − 25 = 100 integers qualify.
Numbers congruent to 2 modulo 5 are 2, 7, 12, 17, 22, ... Testing these modulo 7 gives 2, 0, 5, 3, so 17 is the first that leaves remainder 3. The Chinese remainder theorem guarantees a unique solution modulo 35.
For any two positive integers, the product of the GCD and the LCM equals the product of the numbers. Dividing 1800 by 10 gives an LCM of 180, consistent with gcd(10, 180) = 10 and lcm(10, 180) = 180.
Subtract 7 from both sides to get 3x = 15, then divide both sides by 3 to obtain x = 5. Each step applies the same operation to both sides of the equation to preserve equality.
Distribute the 4 to get 8x - 12, then combine the like terms 8x and 5x to obtain 13x. The constant -12 remains unchanged. Combining like terms requires adding coefficients of the same variable power.
Slope equals the change in y divided by the change in x, so (11 - 3)/(6 - 2) = 8/4 = 2. The slope measures vertical rise per unit of horizontal run.
The expression x^2 - 9 is a difference of squares that factors as (x - 3)(x + 3). Therefore x - 3 is one of its factors. Recognizing the pattern a^2 - b^2 = (a - b)(a + b) is the key step.
Add 5 to both sides to get 2x > 14, then divide by the positive 2 to obtain x > 7. Dividing by a positive number does not reverse the inequality sign.
In slope-intercept form y = mx + b, the value b is the y-intercept. Here b = 8, so the line crosses the y-axis at (0, 8).
Compute each power first: 2^3 = 8 and 3^2 = 9. Adding them gives 17, following the order of operations where exponents precede addition.
Substitute x = 4 into the function to get 2(4) + 1 = 8 + 1 = 9. Function notation f(4) means the output when the input is 4.
We need two numbers that multiply to 6 and add to 5, which are 2 and 3. Thus the factorization is (x + 2)(x + 3). Expanding confirms the middle term is 5x.
Adding the two equations eliminates y, giving 2x = 14, so x = 7. Substituting back yields y = 3, confirming the solution. The elimination method combines equations to remove a variable.
When multiplying powers with the same base, add the exponents: x^3 times x^4 equals x^(3+4) = x^7. The base stays the same while exponents combine.
The quadratic factors as (x - 2)(x - 3) = 0, since -2 and -3 multiply to 6 and add to -5. Setting each factor to zero gives x = 2 and x = 3.
The numerator factors as (x - 2)(x + 2). Canceling the common factor (x - 2) leaves x + 2, valid whenever x is not 2.
Rewrite 16 as 2^4, so 2^(x+1) = 2^4. Equating exponents gives x + 1 = 4, hence x = 3. Matching bases allows direct comparison of exponents.
The denominator cannot equal zero, so x - 4 is not allowed to be zero. This excludes x = 4, leaving all other real numbers in the domain.
Substitute x = 3 and y = 11 into y = 2x + b to get 11 = 6 + b. Solving gives b = 5, which is the y-intercept of the line.
An absolute value equation splits into two cases: x - 3 = 7 and x - 3 = -7. Solving each yields x = 10 and x = -4. Both values satisfy the original equation.
Half of 6 is 3, and 3 squared is 9. Adding and subtracting 9 gives (x^2 + 6x + 9) - 9 + 5 = (x + 3)^2 - 4. Completing the square rewrites the quadratic in vertex form.
Divide coefficients: 2/4 = 1/2. Subtract exponents for like bases: x^(2-1) = x and y^(3-2) = y. The result is (1/2)xy, or (xy)/2.
The numbers are roots of t^2 - 20t + 96 = 0, which factors as (t - 8)(t - 12) = 0. The roots are 8 and 12, so the larger number is 12. Their sum is 20 and product is 96, confirming the answer.
For infinitely many solutions the two equations must be proportional. The second equation divided by 4 gives x + 2y = 3, matching the first after dividing by 2 when k = 1. Thus k = 1 makes the equations equivalent.
Square both sides: (x + 1/x)^2 = x^2 + 2 + 1/x^2 = 9. Subtracting 2 gives x^2 + 1/x^2 = 7. The cross terms always equal 2 in this identity.
Dividing by (x - 2) yields x^2 - 2x - 3, which factors as (x - 3)(x + 1). Thus the full factorization is (x - 2)(x - 3)(x + 1), giving roots 2, 3, and -1.
Rewrite the logarithmic equation in exponential form: x + 3 = 2^4 = 16. Subtracting 3 gives x = 13. Checking, log base 2 of 16 equals 4, confirming the solution.
The critical points are x = 1 (numerator zero) and x = -2 (denominator zero, excluded). Testing intervals shows the expression is nonnegative when x is at most -2 or at least 1, with x = -2 excluded.
The slope m is the change in y divided by the change in x: m = (11 - 3) / (6 - 2) = 8 / 4 = 2. This measures the steepness of the line.
Apply the power of a product rule: (ab)^n = a^n b^n. Also use (x^m)^n = x^{mn}. So (3x^2y)^3 = 3^3 (x^2)^3 y^3 = 27x^6y^3.
Use the point-slope form: y - y1 = m(x - x1). Substituting m = -2 and (1, 4) gives y - 4 = -2(x - 1). Simplify to y = -2x + 6.
Compute factorials: 5! = 120, 3! = 6, and 2! = 2. Then 120 / (6 * 2) = 120 / 12 = 10. This is also the binomial coefficient C(5,2).
Add 5 to both sides to get 3x < 12. Then divide both sides by 3, which is positive, so the inequality sign stays the same: x < 4.