question stringlengths 12 86 | problem stringclasses 31
values | how_to_solve stringlengths 8 96 | answer stringlengths 1 24 |
|---|---|---|---|
What is -45 mod 19 (remainder after division)? | Compute remainder when a is divided by b. | Divide -45 by 19 and take the integer remainder 0..18. | 12 |
Solve: |3x + -10| = 9 | Find integer solutions x, if any. | Set inside equal to ±c and solve both linear equations. | no integer solutions |
Solve for x: 4^x = 4 | Solve the exponential equation for integer x. | Find x such that repeated multiplication of 4 yields 4. | 1 |
What is 20% of 45? | Compute the percent value (integer-safe). | Compute 20/100 × 45. | 9 |
Solve for x: 3(x + -1) = 9 | Solve a two-step equation. | Divide both sides by a, then subtract b. | 4 |
Solve: |-2x + 18| ≤ 7 | Find integer range of x satisfying the absolute inequality. | Isolate and produce interval then take integer bounds. | 6 ≤ x ≤ 12 |
What is 189 ÷ -21? | Divide evenly. | Divide 189 by -21. | -9 |
Solve: |-4x + 18| ≤ 11 | Find integer range of x satisfying the absolute inequality. | Isolate and produce interval then take integer bounds. | 2 ≤ x ≤ 7 |
What is 50 × 43? | Multiply the numbers. | Multiply 50 by 43. | 2150 |
Solve for x: -2x^2 + -2x + 12 = 0 | Quadratic with integer roots and non-unit leading coefficient. | Factor or use quadratic formula; roots are integers. | 2, -3 |
Solve for x: 2x^2 - 18x + 16 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | 8, 1 |
Solve the system:
-2x + 1y + 1z = -19
-2x + 0y + 3z = -26
-1x + 3y + 1z = -25 | Find integer x,y,z satisfying the linear system. | Use elimination or Cramer's rule (matrix is invertible). | x=4, y=-5, z=-6 |
What is 0 × -27? | Multiply the numbers. | Multiply 0 by -27. | 0 |
Solve for x: (8/7)x + -11 = -51 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | -35 |
What is 7^1? | Evaluate the integer exponent. | Multiply 7 by itself 1 times (0th power = 1). | 7 |
What is -360 ÷ 20? | Divide evenly (negative allowed). | Divide -360 by 20. | -18 |
Solve for x: 2x + -1 = 21 | Solve the linear equation. | Subtract b from both sides, then divide by a. | 11 |
What is 192 ÷ -12? | Divide evenly (negative allowed). | Divide 192 by -12. | -16 |
Solve for x: log base 3 of x = 6 | Compute x from the log equation (integer-safe). | x = 3^6. | 729 |
Solve for x: (6x + -8)/(6x + -8) = 1 | Rational equation that reduces to a linear equation. | Cross-multiply to get linear equation and solve for x (ensure denominator ≠ 0). | -6 |
Find smallest non-negative x such that -2x ≡ 4 (mod 30) | Solve linear congruence. | Use gcd and modular inverse on reduced modulus (if solution exists). | 13 |
Solve: |-2x + 4| = 18 | Find integer solutions x, if any. | Set inside equal to ±c and solve both linear equations. | -7, 11 |
What is 134 + 281? | Add the two numbers. | Add 134 and 281. | 415 |
Convert the improper fraction 146/7 to a mixed number. | Convert to mixed number and simplify the fractional part. | Divide numerator by denominator to get whole part and remainder; simplify remainder/denominator. | 20 6/7 |
Factor the polynomial: 1x^2 + 8x + 12 | Factor quadratic into integer linear factors. | Find integers r1 and r2 such that polynomial = (x - r1)(x - r2). | (x - -6)(x - -2) |
What is -76 - 162? | Subtract the second number from the first. | Subtract 162 from -76. | -238 |
Solve for x: (3/1)x + 7 = -14 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | -7 |
Solve for x: -2x^2 - 12x - 10 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | -5, -1 |
Solve the system:
-4x + -3y + -1z = -3
-4x + -3y + -3z = -5
-1x + -4y + -2z = 17 | Find integer x,y,z satisfying the linear system. | Use elimination or Cramer's rule (matrix is invertible). | x=5, y=-6, z=1 |
What is 48 ÷ 4? | Divide evenly (negative allowed). | Divide 48 by 4. | 12 |
Solve the system:
0x + 4y + 3z = -9
2x + 3y + -1z = -12
2x + 1y + -1z = -6 | Find integer x,y,z satisfying the linear system. | Use elimination or Cramer's rule (matrix is invertible). | x=-1, y=-3, z=1 |
What is -158 + -220? | Add the two numbers. | Add -158 and -220. | -378 |
Solve: 10x + -7 ≤ -37 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x ≤ -3 |
Simplify: -7x + 8x + 2y | Combine like terms. | Add coefficients of like terms (x with x, y with y). | 1x + 2y |
Solve for x: -2x^2 - 2x + 24 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | -4, 3 |
What is -180 ÷ -12? | Divide evenly. | Divide -180 by -12. | 15 |
Solve: -6x + -4 < -82 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x < 13 |
What is 12 + 51? | Add the two numbers. | Add 12 and 51. | 63 |
Solve the system:
-2x + -4y + -4z = -30
3x + 0y + 1z = 15
-3x + -1y + -2z = -20 | Find integer x,y,z satisfying the linear system. | Use elimination or Cramer's rule (matrix is invertible). | x=5, y=5, z=0 |
Simplify: -1x + -2x + 6y | Combine like terms. | Add coefficients of like terms (x with x, y with y). | -3x + 6y |
Solve for x: (-9/2)x + 4 = -122 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | 28 |
Convert the improper fraction -18/5 to a mixed number. | Convert to mixed number and simplify the fractional part. | Divide numerator by denominator to get whole part and remainder; simplify remainder/denominator. | -3 -3/5 |
Solve: 6x + 4 ≤ 58 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x ≤ 9 |
What is 4 ÷ 4? | Divide evenly (negative allowed). | Divide 4 by 4. | 1 |
Solve for x: 3x^2 + -15x + -18 = 0 | Quadratic with integer roots and non-unit leading coefficient. | Factor or use quadratic formula; roots are integers. | -1, 6 |
Solve for x: (7/10)x + -16 = 40 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | 80 |
Solve for x: (-7/12)x + -17 = -66 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | 84 |
Solve for x: 1x^3 + 3x^2 + -9x + 5 = 0 | Cubic polynomial factorable to (x - r1)(x - r2)(x - r3). | Find integer root by rational root theorem then factor. | -5, 1, 1 |
Solve: |1x + 2| = 5 | Find integer solutions x, if any. | Set inside equal to ±c and solve both linear equations. | 3, -7 |
What is 100 ÷ 10? | Divide evenly. | Divide 100 by 10. | 10 |
Solve the system:
0x + 2y + -2z = -22
3x + 4y + 2z = 1
4x + 1y + -1z = 1 | Find integer x,y,z satisfying the linear system. | Use elimination or Cramer's rule (matrix is invertible). | x=3, y=-5, z=6 |
Solve for x: sqrt(-6x + 120) = 12 | Radical equation; return integer solution if it exists. | Square both sides to get ax + b = c^2, then solve. | -4 |
Solve the system:
4x + -4y = 0
-2x + 3y = 0 | Find integer x and y that satisfy both equations. | Use substitution or elimination. | x=0, y=0 |
What is 46% of 200? | Compute the percent value (integer-safe). | Compute 46/100 × 200. | 92 |
Convert the improper fraction -19/8 to a mixed number. | Convert to mixed number and simplify the fractional part. | Divide numerator by denominator to get whole part and remainder; simplify remainder/denominator. | -2 -3/8 |
Solve for x: 4x^2 + 0x + -36 = 0 | Quadratic with integer roots and non-unit leading coefficient. | Factor or use quadratic formula; roots are integers. | -3, 3 |
Solve the system:
2x + -4y + 0z = 26
0x + -3y + 1z = 24
-1x + 3y + 1z = -13 | Find integer x,y,z satisfying the linear system. | Use elimination or Cramer's rule (matrix is invertible). | x=1, y=-6, z=6 |
Simplify -60/12 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | -5 |
What is -129 - -105? | Subtract the second number from the first. | Subtract -105 from -129. | -24 |
What is 384 ÷ 24? | Divide evenly. | Divide 384 by 24. | 16 |
Simplify 30/3 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | 10 |
Solve the system:
-3x + 3y = -6
1x + -5y = 10 | Find integer x and y that satisfy both equations. | Use substitution or elimination. | x=0, y=-2 |
What is 48 × 56? | Multiply the numbers. | Multiply 48 by 56. | 2688 |
Solve the system:
-3x + 3y + 0z = 9
1x + 3y + 2z = -1
-3x + 1y + -4z = 17 | Find integer x,y,z satisfying the linear system. | Use elimination or Cramer's rule (matrix is invertible). | x=-1, y=2, z=-3 |
Solve for x: 10x + 28 = -42 | Solve the linear equation. | Subtract b from both sides, then divide by a. | -7 |
Solve for x: (-6x + 0)/(3x + 6) = -1 | Rational equation that reduces to a linear equation. | Cross-multiply to get linear equation and solve for x (ensure denominator ≠ 0). | 2 |
Convert the improper fraction -31/3 to a mixed number. | Convert to mixed number and simplify the fractional part. | Divide numerator by denominator to get whole part and remainder; simplify remainder/denominator. | -10 -1/3 |
What is 18 × 2? | Multiply the numbers. | Multiply 18 by 2. | 36 |
Solve for x: (-1/7)x + -11 = -5 | Linear equation with fractional coefficient. | Subtract c then multiply by b/a (watch sign). | -42 |
What is 29 + -11? | Add the two numbers. | Add 29 and -11. | 18 |
Simplify: 7x + 7x + -7y | Combine like terms. | Add coefficients of like terms (x with x, y with y). | 14x - 7y |
Simplify 32/16 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | 2 |
Solve: -3x + -19 < 2 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x < -7 |
Solve: -7x + -9 > -93 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x > 12 |
What is -90 ÷ -10? | Divide evenly. | Divide -90 by -10. | 9 |
Solve: |-7x + 10| = 20 | Find integer solutions x, if any. | Set inside equal to ±c and solve both linear equations. | no integer solutions |
Simplify: 2x + -6x + 3y | Combine like terms. | Add coefficients of like terms (x with x, y with y). | -4x + 3y |
Simplify 126/14 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | 9 |
Solve the system:
0x + 3y + -3z = -15
-4x + 3y + -1z = -41
-4x + 2y + -1z = -35 | Find integer x,y,z satisfying the linear system. | Use elimination or Cramer's rule (matrix is invertible). | x=6, y=-6, z=-1 |
Simplify 120/15 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | 8 |
Simplify: -4x + 3x + -5y | Combine like terms. | Add coefficients of like terms (x with x, y with y). | -1x - 5y |
Solve for x: sqrt(2x + -14) = 2 | Radical equation; return integer solution if it exists. | Square both sides to get ax + b = c^2, then solve. | 9 |
Solve the system:
1x + -2y + -3z = -12
-4x + -4y + 3z = -18
2x + 2y + 4z = 20 | Find integer x,y,z satisfying the linear system. | Use elimination or Cramer's rule (matrix is invertible). | x=2, y=4, z=2 |
Solve for x: 3(x + 4) = -3 | Solve a two-step equation. | Divide both sides by a, then subtract b. | -5 |
Solve for x: sqrt(-5x + -26) = 2 | Radical equation; return integer solution if it exists. | Square both sides to get ax + b = c^2, then solve. | -6 |
Simplify -66/6 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | -11 |
Solve for x: 5^x = 1 | Solve the exponential equation for integer x. | Find x such that repeated multiplication of 5 yields 1. | 0 |
What is 2^0? | Evaluate the integer exponent. | Multiply 2 by itself 0 times (0th power = 1). | 1 |
Simplify: 1x + 4x + 4y | Combine like terms. | Add coefficients of like terms (x with x, y with y). | 5x + 4y |
What is -21 ÷ 3? | Divide evenly. | Divide -21 by 3. | -7 |
Solve the system:
5x + 4y = 76
-1x + -5y = -53 | Find integer x and y that satisfy both equations. | Use substitution or elimination. | x=8, y=9 |
Find one integer solution to 2x + 8y = 34 | Linear Diophantine equation (return one integer pair). | Find any particular solution (here constructed). | x=5, y=3 |
What is -245 + 191? | Add the two numbers. | Add -245 and 191. | -54 |
Simplify -49/7 | Reduce the fraction to simplest form. | Divide numerator and denominator by their GCD and normalize sign. | -7 |
Solve: 5x + -20 > -120 | Solve the inequality. | Isolate x by subtracting b then dividing by a (flip sign if dividing by negative). | x > -20 |
Solve for x: log base 2 of x = 6 | Compute x from the log equation (integer-safe). | x = 2^6. | 64 |
Solve for x: (8x + -6)/(2x + -7) = 26 | Rational equation that reduces to a linear equation. | Cross-multiply to get linear equation and solve for x (ensure denominator ≠ 0). | 4 |
Simplify: 0x + 2x + 3y | Combine like terms. | Add coefficients of like terms (x with x, y with y). | 2x + 3y |
Solve for x: 2x^2 - 4x - 16 = 0 | Solve the quadratic equation with integer roots. | Factor or use quadratic formula (should factor cleanly). | -2, 4 |
Find one integer solution to 7x + 9y = 116 | Linear Diophantine equation (return one integer pair). | Find any particular solution (here constructed). | x=5, y=9 |
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