How this instrument works
A multiplicative inverse of a number, taken modulo a prime, is whatever value multiplies against the original and lands on a remainder of exactly 1. Ordinary division carries no meaning inside modular arithmetic, so this stand-in value is what makes canceling a multiplication possible once every number gets reduced by that same modulus. For 7 modulo 11, the inverse turns out to be 8: multiply them together, 7×8=56, and 56 divided by 11 leaves a remainder of 1 — 8 is exactly the value that cancels a multiplication by 7 back down to 1 within that modulus.
Fermat's Little Theorem supplies a direct route to that value whenever the modulus is prime: for any number not divisible by the modulus, raising it to one less than the modulus always reduces to 1. Splitting that exponent as 1 plus (modulus minus 2) turns the identity into a product — the original number times itself raised to (modulus minus 2) also reduces to 1 — which means that single power, once reduced, already is the inverse being sought. One exponentiation and one remainder replace what would otherwise need a coefficient-tracking loop.
This site already carries a plain Inverse Modulo sheet built on that identical rearrangement of Fermat's Little Theorem, and this page does not pretend its underlying formula differs — it doesn't. What earns this sheet its own place is the setting: computing a modular inverse this way is exactly the operation RSA-style key generation performs internally, deriving a private exponent from a public one against a totient built from two enormous primes, rather than the small teaching values used below.
Public-key cryptography leans on this arithmetic constantly, since a modular inverse lets one party undo an operation performed by someone else without either side ever exposing a private key directly. Swap the small prime 11 for one hundreds of digits long, and nothing about the method itself changes — only the size of the numbers being raised and reduced grows, while the formula stays the exact same single line.
- Enter the number you want to invert into Number (a).
- Enter a prime into Prime modulus (p) — the Fermat shortcut only holds when this value is genuinely prime.
- Read Multiplicative inverse of a mod p for the value Fermat's Little Theorem produces directly.
- Multiply your entered a by the returned inverse and reduce by your entered p yourself, confirming the product lands on a remainder of 1.
- Change either a or the prime modulus and recompute to see how the inverse shifts with each new pair.
Worked example — inverting 7, modulo the prime 11
Set Number (a) to 7 and Prime modulus (p) to 11. Fermat's Little Theorem calls for 7 raised to the ninth power, then reduced by 11 — and repeated squaring keeps that manageable without ever forming the full nine-digit value of that power directly. Square once: 7²=49, which reduces to 5. Square again: 5²=25, reducing to 3. Square once more: 3²=9. Multiply that by one final factor of 7: 9×7=63, and 63 reduces to 8 — so Multiplicative inverse of a mod p reads 8, matching the shortcut's promise exactly.
Multiplying back confirms it without needing any of the exponent arithmetic: 7×8=56, and 56 is five elevens plus 1, a remainder of exactly 1. The same pattern holds at smaller scale too — inverting 3 modulo the prime 7 gives 5, since 3×5=15 and 15 is one more than 14, itself two sevens. Real RSA key generation runs this identical remainder-and-multiply check internally, only against primes hundreds of digits long instead of 11 or 7.
Questions
What is a multiplicative inverse modulo a prime?
It is the number that, multiplied by the original value and reduced by that prime, leaves a remainder of exactly 1 — the modular substitute for division. For 7 modulo 11, the inverse is 8, since 7×8=56 and 56 divided by 11 leaves remainder 1.
Why does this page use the same formula as the existing Inverse Modulo calculator?
Because the underlying math genuinely is the same — raising a number to two less than the modulus and reducing by it, straight from Fermat's Little Theorem, is the correct shortcut whenever that modulus is prime, and this page does not dress it up as something new. What differs is the framing: this sheet highlights how the identical computation sits inside RSA-style key generation, where a private exponent gets derived from a public one through a modular inverse.
How does this connect to RSA encryption?
RSA key generation needs a private exponent that is the modular inverse of the public exponent, taken against a totient built from two enormous primes. That arithmetic is the exact operation this sheet performs — raise a number to two less than the modulus and reduce — just carried out with primes hundreds of digits long rather than the small teaching values of 7, 11, 3, and 5 used above.
Why must the modulus be prime for this shortcut to work?
Fermat's Little Theorem, which the formula rearranges, is only stated for a prime modulus. Under a composite one, some numbers share a common factor with it and have no inverse whatsoever, so the underlying identity breaks down and a different technique, such as the extended Euclidean algorithm, becomes necessary instead.
What happens if the entered number is a multiple of the modulus?
No inverse exists, because a multiple of the modulus is congruent to zero once reduced, and nothing multiplied by zero can ever leave a remainder of 1. This sheet checks for that case and flags it rather than returning a meaningless answer.
Does using a much larger prime change how the computation works?
Only in the size of the numbers involved, not in the number of steps: raise the value, reduce by the modulus, done. Repeated squaring, the same shortcut used in the worked example above, keeps that practical even when the modulus runs into the hundreds of digits real cryptographic systems rely on.