Fraction, Mixed Number, GCF, LCM & Prime Factorization Calculator

Perform exact rational fraction arithmetic and compute GCF, LCM, long division remainder, and prime factors.

Exact Simplified Fraction & Mixed Number
19/12 (1.583333)
Mixed: 1 7/12 • LCD of denominators (4, 6) = 12
Quotient & Remainder Q = 1, R = 7
GCF & LCM of (4, 6) GCF = 2, LCM = 12
Prime Factors of 24 2 × 2 × 2 × 3
Decimal Equivalent 1.583333
Inputs
Top integer of first fraction (a/b).
Bottom integer of first fraction (also used for GCF/LCM).
Operation applied between (a/b) and (c/d).
Top integer of second fraction (c/d).
Bottom integer of second fraction (also used for GCF/LCM).
Results & Graph
Exact Simplified Fraction & Mixed Number
19/12 (1.583333)
Mixed: 1 7/12 • LCD of denominators (4, 6) = 12
Quotient & Remainder
Q = 1, R = 7
GCF & LCM of (4, 6)
GCF = 2, LCM = 12
Prime Factors of 24
2 × 2 × 2 × 3
Decimal Equivalent
1.583333

Decimal Magnitude Comparison

First Fraction vs. Second Fraction vs. Result
Graph 1 • Partitioned Fraction Grid Bars Shaded numerator cells out of denominator partitions
Fraction 1 (3/4)0.7500 Fraction 2 (5/6)0.8333 Result (1.583)1.5833
Graph 2 • Exact Coordinate Placement on the Real Number Line Plots Fraction 1, Fraction 2, and Combined Result
3/4 (0.75) 5/6 (0.83) Result = 1.5833 0 1.0 2
Graph 3 • Fraction Magnitudes & Common Denominator Scale Reduced via Euclidean Greatest Common Divisor (GCD)
First (3/4) 0.75 Second (5/6) 0.833 Result (19/12) 1.583
Summary: Simplified fraction is 19/12 (1.5833), GCF is 2, and LCM is 12.
Summary: Evaluating (3/4) and (5/6) yields 38/24, which reduces via GCF (2) to 19/12 (decimal 1.583333). For the denominators 4 and 6, the Greatest Common Factor is 2 and Least Common Multiple (LCD) is 12.
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Formula & Mathematical Methodology: Rational Arithmetic & Euclidean GCD / LCM Identities

ab ± cd = ad ± bcbd,   lcm(b, d) = |b · d|gcd(b, d)
a/b ± c/d = (ad ± bc) / bd; gcd(x, y) via Euclid; lcm(b, d) = |b × d| / gcd(b, d)
  • gcd(x, y): Greatest Common Divisor (Factor) of integers x and y
  • lcm(b, d): Least Common Multiple (Least Common Denominator)

Rounding: Exact integer rational reduction; decimal readout rounded to 6 places.

Worked Example: Add 3/4 + 5/6 and Find GCF & LCM of 4 and 6

Compute 3/4 + 5/6 in simplest improper and mixed-number form.

  1. GCF(4, 6) = 2; Least Common Denominator LCM(4, 6) = (4 × 6) ÷ 2 = 12.
  2. 3/4 + 5/6 = 9/12 + 10/12 = 19/12.
  3. Long division: 19 ÷ 12 = Quotient 1, Remainder 7 → Mixed number 1 7/12 (1.583333).
Result: 19/12 = 1 7/12 (1.583333); GCF(4,6) = 2, LCM(4,6) = 12.

Frequently Asked Questions

How are GCF and LCM mathematically connected?

For any two positive integers a and b, the product of their Greatest Common Factor and Least Common Multiple equals the product of the numbers themselves: GCF(a, b) × LCM(a, b) = a × b.

Mathematics & Statistics · Rational Arithmetic & Number Theory

Fraction, Mixed Number, GCF, LCM & Prime Factorization Calculator: inputs, method, and how to read the result

Perform exact rational fraction arithmetic and compute GCF, LCM, long division remainder, and prime factors.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • First Fraction Numerator (a): Top integer of first fraction (a/b).
  • First Fraction Denominator (b): Bottom integer of first fraction (also used for GCF/LCM).
  • Arithmetic Operation: Operation applied between (a/b) and (c/d).
  • Second Fraction Numerator (c): Top integer of second fraction (c/d).

What the calculation includes

With the default values shown, Exact Simplified Fraction & Mixed Number is 19/12 (1.5833). The calculation follows Rational Arithmetic & Euclidean GCD / LCM Identities; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing First Fraction Numerator (a) from 3 to 4 changes Exact Simplified Fraction & Mixed Number from 19/12 (1.5833) to 11/6 (1.8333). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Inputs are truncated to integers for exact rational and prime factorization arithmetic.
  • Prime factorization is capped at standard safe integers.

Scope: Global Check the assumptions and limitations against your situation before relying on the estimate.

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