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Primes and factors

Is 221 prime? The smallest prime factor of 391? The gcd and lcm of 84 and 126? Numbers built to look prime, and magnitudes that never stop growing.

  • Number sense
  • Timed
  • Endless levels

How it plays

Four kinds of question: whether a number is prime; its smallest prime factor; the greatest common divisor of two numbers; their least common multiple.

The composites are chosen to look prime — odd, digit sum not a multiple of 3, not ending in 5 — like 91 = 7 × 13 and 221 = 13 × 17. A number is prime when no prime up to its square root divides it. For the gcd, Euclid: divide, keep the remainder, repeat; the lcm is the product divided by the gcd.

Levels grow the numbers without end and shorten the time. The working shows the factorisation, the square-root bound, or Euclid step by step.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This trains factor sense — spotting primes, divisors and common factors. It is practice at this, nothing wider.

Learn

How to get better at primes and factors: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Run the free tests first: even, digit sum a multiple of 3, ending in 0 or 5. The numbers here are built to pass them, so expect them to tell you nothing — and move on fast.
  2. Find the square root roughly: 221 is under 15² = 225, so only primes up to 13 need trying. A composite always has a factor no bigger than its square root.
  3. Try 7, 11, 13, 17 … in order, with their quick tests, and stop at the first that divides. For the smallest prime factor, that first one is the answer.
  4. For a gcd, use Euclid: replace the larger number by its remainder on division by the smaller, and repeat until it divides exactly. For an lcm, divide one number by the gcd and multiply by the other.

Techniques and tricks

  • Stop at the square root

    If n = a × b, the smaller of a and b is at most √n. So once you pass √n with no factor found, n is prime.

    e.g. 211: √211 is just over 14, and 7, 11 and 13 all leave remainders, so 211 is prime

  • Know the numbers that look prime

    Learn the odd composites with no factor of 3 or 5 up to 300. They come up again and again.

    e.g. 91 = 7 × 13, 119 = 7 × 17, 133 = 7 × 19, 143 = 11 × 13, 161 = 7 × 23, 187 = 11 × 17, 209 = 11 × 19, 221 = 13 × 17, 247 = 13 × 19, 289 = 17²

  • Difference of two squares

    If n is just below a square, try n = a² − b² = (a − b)(a + b). It finds factors near the square root in one step.

    e.g. 391 = 400 − 9 = 20² − 3² = 17 × 23; 221 = 225 − 4 = 15² − 2² = 13 × 17

  • Quick tests for 7, 13 and 17

    Drop the last digit and fold it back: for 7, subtract twice it; for 13, add four times it; for 17, subtract five times it. The result divides exactly when the number does.

    e.g. 371 → 37 − 2 = 35, so 7 divides it. 221 → 22 + 4 = 26, so 13 does. 391 → 39 − 5 = 34, so 17 does.

  • The gcd divides the difference

    Any common divisor of two numbers also divides their difference. Subtracting is Euclid’s algorithm in its simplest form, and often finishes it in one step.

    e.g. 84 and 126: 126 − 84 = 42, and 42 divides 84, so the gcd is 42

  • Divide before you multiply

    lcm(a, b) = a × b ÷ gcd(a, b). Divide one number by the gcd first, so the numbers stay small.

    e.g. lcm(84, 126): 84 ÷ 42 = 2, and 2 × 126 = 252

As the levels climb

  • All four kinds are open from the start; what climbs is size. The numbers to test gain a digit every couple of steps, so a bigger range of primes up to the square root has to be tried. Learn the squares of 11 to 31 to find that bound instantly.
  • For the smallest prime factor, the factor itself grows: 3 and 5 are allowed only at first, then it is always 7 or more. Go straight to 7 and climb; the 2, 3 and 5 tests will not find it.
  • Gcd and lcm questions grow their common factor and the numbers around it. Euclid takes more steps; write each remainder as you go and you will not lose your place.
  • Between the size steps the clock shrinks to 62% of its starting time. The time grows with the length of the number, so a six-digit number gets longer than a three-digit one — but not long enough for trial division by every prime. Lean on the squares.

Traps to avoid

  • Calling a number prime because it passed the tests for 2, 3 and 5. Every composite here is built from primes of 7 or more for exactly that reason; 91, 221 and 391 all look prime.
  • The larger factor given for the smallest one. 391 = 17 × 23, and 23 is among the options; the question asks for 17.
  • The prime next to the right one. When the smallest factor is 17, 13 and 19 turn up among the options. Check by dividing, not by feel.
  • For an lcm, the plain product a × b. It counts the common factor twice: lcm(84, 126) is 252, not 10,584.
  • For a gcd, the difference or a part of the answer. The difference 126 − 84 = 42 happens to be the gcd here, but often it is only a multiple of it: for 231 and 363 the difference is 132 and the gcd is 33.

How to practise

  • Use “Questions” to drill “Is it prime?” on its own for a while: it is the quickest way to learn the look-alike composites by heart.
  • Switch “Answer by” to “Typing the number” for “Smallest prime factor”, “Greatest common divisor” and “Least common multiple” once you choose well; “Is it prime?” stays a choice between Prime and Not prime.
  • Raise “Options shown” towards 8 so the neighbouring primes and the larger factor are always on offer; then only a real division picks the answer.

Levels

Each question has a clock that comes from the work it asks for. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More number sense

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