Prime Factorization and Divisibility Calculator

ANA›Life Services Authority›National Calculator Authority›Prime Factorization and Divisibility Calculator

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Prime Factorization and Divisibility Calculator

Enter a positive integer to find its prime factorization, divisors, GCD, and LCM with a second number.

Primary Number (N)

Second Number (M) (optional – for GCD & LCM)

Calculate

### Results for N =

Prime Factorization

Exponential Form

Number of Divisors τ(N)

Sum of Divisors σ(N)

All Divisors

Is Perfect Number?

Is Prime?

#### Divisibility Rules for N

Divisor Divisible? Rule

#### GCD & LCM with M =

GCD(N, M)

LCM(N, M)

Are they Coprime?

function priCalc() { const errEl = document.getElementById('pri-error'); const resEl = document.getElementById('pri-result'); errEl.style.display = 'none'; resEl.style.display = 'none';

// --- Input validation --- const rawN = document.getElementById('pri-num1').value.trim(); const rawM = document.getElementById('pri-num2').value.trim();

if (rawN === '') { errEl.textContent = 'Please enter a Primary Number N.'; errEl.style.display = 'block'; return; }

const N = parseInt(rawN, 10); if (!Number.isInteger(N) || N 1e9) { errEl.textContent = 'N must be a positive integer between 1 and 1,000,000,000.'; errEl.style.display = 'block'; return; }

let M = null; if (rawM !== '') { M = parseInt(rawM, 10); if (!Number.isInteger(M) || M 1e9) { errEl.textContent = 'M must be a positive integer between 1 and 1,000,000,000.'; errEl.style.display = 'block'; return; } }

// ── Prime Factorization ────────────────────────────────────────────────── // Trial division: N = p1^e1 × p2^e2 × … function primeFactors(n) { const factors = {}; let d = 2; while (d * d 1) factors[n] = (factors[n] || 0) + 1; return factors; }

const factors = primeFactors(N); const primes = Object.keys(factors).map(Number).sort((a,b) => a-b);

// Factorization string: 2 × 2 × 3 … const factList = []; primes.forEach(p => { for (let i = 0; i factors[p] > 1 ? ${p}^${factors[p]} : ${p}).join(' × ') || ${N};

// ── Number of Divisors: τ(N) = ∏(eᵢ + 1) ─────────────────────────────── let tau = 1; primes.forEach(p => { tau *= (factors[p] + 1); });

// ── Sum of Divisors: σ(N) = ∏ (p^(e+1) − 1)/(p − 1) ─────────────────── let sigma = 1; primes.forEach(p => { const e = factors[p]; sigma *= (Math.pow(p, e + 1) - 1) / (p - 1); }); sigma = Math.round(sigma);

// ── All Divisors (only list if τ ≤ 200 to keep UI clean) ──────────────── let divisors = []; if (tau a-b); }

// ── Perfect Number: σ(N) = 2N ─────────────────────────────────────────── const isPerfect = (sigma === 2 * N);

// ── Is Prime ───────────────────────────────────────────────────────────── const isPrime = (primes.length === 1 && factors[primes[0]] === 1);

// ── Divisibility Rules ─────────────────────────────────────────────────── // Rules for divisors 2–13 const divRules = [ { d: 2, check: n => n % 2 === 0, rule: 'Last digit is even (0,2,4,6,8).' }, { d: 3, check: n => { let s=0; String(n).split('').forEach(c=>s+=+c); return s%3===0; }, rule: 'Sum of digits is divisible by 3.' }, { d: 4, check: n => n % 4 === 0, rule: 'Last two digits form a number divisible by 4.' }, { d: 5, check: n => n % 5 === 0, rule: 'Last digit is 0 or 5.' }, { d: 6, check: n => n % 6 === 0, rule: 'Divisible by both 2 and 3.' }, { d: 7, check: n => n % 7 === 0, rule: 'Double last digit, subtract from rest; repeat until small.' }, { d: 8, check: n => n % 8 === 0, rule: 'Last three digits form a number divisible by 8.' }, { d: 9, check: n => { let s=0; String(n).split('').forEach(c=>s+=+c); return s%9===0; }, rule: 'Sum of digits is divisible by 9.' }, { d: 10, check: n => n % 10 === 0, rule: 'Last digit is 0.' }, { d: 11, check: n => { let alt=0, s=String(n); for(let i=0;i n % 12 === 0, rule: 'Divisible by both 3 and 4.' }, { d: 13, check: n => n % 13 === 0, rule: 'Add 4× last digit to rest; repeat until small.' }, ];

// ── GCD (Euclidean algorithm) & LCM ───────────────────────────────────── function gcd(a, b) { while (b) { [a, b] = [b, a % b]; } return a; } function lcm(a, b) { return (a / gcd(a, b)) * b; }

// ── Populate DOM ───────────────────────────────────────────────────────── document.getElementById('pri-n-display').textContent = N.toLocaleString(); document.getElementById('pri-factorization').textContent = factStr; document.getElementById('pri-exp-form').textContent = expStr; document.getElementById('pri-num-divisors').textContent = tau.toLocaleString(); document.getElementById('pri-sum-divisors').textContent = sigma.toLocaleString(); document.getElementById('pri-all-divisors').textContent = tau { const yes = check(N); const tr = document.createElement('tr'); tr.innerHTML = ${d} ${yes?'✅ Yes':'❌ No'} ${rule}; if (tbody.children.length % 2 === 1) tr.style.background = '#f4f6fa'; tbody.appendChild(tr); });

// GCD / LCM section const gcdSection = document.getElementById('pri-gcd-section'); if (M !== null) { const g = gcd(N, M); const l = lcm(N, M); document.getElementById('pri-m-display').textContent = M.toLocaleString(); document.getElementById('pri-gcd').textContent = g.toLocaleString(); document.getElementById('pri-lcm').textContent = l.toLocaleString(); document.getElementById('pri-coprime').textContent = g === 1 ? '✅ Yes (GCD = 1)' : '❌ No'; gcdSection.style.display = 'block'; } else { gcdSection.style.display = 'none'; }

resEl.style.display = 'block'; }

#### Formulas Used

Prime Factorization (Trial Division): Divide N by every integer d from 2 up to √N. Each time d divides N, record it and divide N by d. Any remainder > 1 is itself prime. N = p₁^e₁ × p₂^e₂ × … × pₖ^eₖ

Number of Divisors: τ(N) = (e₁ + 1)(e₂ + 1) … (eₖ + 1)

Sum of Divisors: σ(N) = ∏ᵢ (pᵢ^(eᵢ+1) − 1) / (pᵢ − 1)

Perfect Number: N is perfect when σ(N) = 2N (i.e., the sum of proper divisors equals N).

GCD (Euclidean Algorithm): gcd(a, b) = gcd(b, a mod b), base case gcd(a, 0) = a

LCM: lcm(N, M) = (N × M) / gcd(N, M)

#### Assumptions & References

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References

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