If r is the remainder when 80 is divided by 9 and n is the remainder when r is divided by 3, what is the value of rn?

If r is the remainder when 80 is divided by 9 and n is the remainder when r is divided by 3, what is the value of rn?

unknown values: x

equation: r = remainder(80, 9)

equation: n = remainder(r, 3)

equation: x = rn

answer: 16

The two digits in the numerator of a fraction are reversed in its denominator. If 1 is subtracted from both the numerator and denominator, the value of the resulting fraction is 1/2. The fraction whose numerator is the difference and whose denominator is the sum of the units and tens digits equals 2/5. Find the original fraction.

unknown values: m

equation: m = (10*x+y)/(10*y+x)

equation: (10*x+y-1)/(10*y+x-1) = 1/2

equation: (y-x)/(x+y) = 2/5

answer: 37/73

The numerator of a fraction is 7 less than the denominator, and if 4 is subtracted from the numerator while 1 is added to the denominator, the resulting fraction is equivalent to 1/3. Find the numerator and denominator of the original fraction.

unknown values: x, y

equation: x = y - 7

equation: (x - 4) / (y + 1) = 1/3

answer: 10; 17

The difference between the product and the sum of two different integers is equal to the sum of their GCD (greatest common divisor) and LCM (least common multiple). Find all these pairs of numbers.

unknown values: m,n

equation: mn - (m+n) = gcd(m,n) + lcm(m,n)

equation: is_integer(m)

equation: is_integer(n)

answer: {4; 6} or {3; 6}

twelve divided by the sum of x and 2 equals the quotient of 4 and the difference of x and 2. Find x.

unknown values: x

equation: 12/(x+2) = 4/(x-2)

answer: 4

the LCM of two numbers is 200 and their GCF is 10. The sum of the numbers is 90. Find the numbers.

unknown values: x,y

equation: lcm(x,y) = 200

equation: gcd(x,y) = 10

equation: x + y = 90

answer: {40; 50}

find two numbers such that their product, their sum, and their difference have the ratio 5:3:2.

unknown values: m,n

equation: mn / (m+n) = 5/3

equation: (m + n) / (m - n) = 3/2

answer: {10; 2}

Find the value of x which satisfies the equation e^(2x) + e^x - 6 = 0

unknown values: x

equation: e^(2*x) + e^x - 6 = 0

answer: ln2 | 0.693

Take 1 less than the square of the sixth prime number and subtract this from the least common multiple of 44 and 54. What is the result?

unknown values: r

equation: r = lcm(44, 54) - (13^2 - 1)

answer: 1020

Find the number of terms in the following geometric sequence:-409.6, 102.4, -25.6,..., 0.025

unknown values: n

equation: 102.4 = -409.6*r

equation: 0.025 = -409.6 * r^(n-1)

answer: 8