Automorphic Number Checker
Test if a number is automorphic.
Result
How it is calculated
The tool squares the whole number n, writes the square out as text and compares its last digits with n written as text. If the final digits of the square are identical to n, the number is automorphic. The number of digits compared is the number of digits of n itself, so a three-digit number is compared with the last three digits of its square. The tool shows the square so you can see the comparison. Whole numbers from 0 to 1,000,000 are accepted; decimals are cut to the integer part, and negative numbers or numbers above one million return no result. The numbers 0 and 1 count as automorphic, since 0² = 0 and 1² = 1. Equivalent condition: n² − n must be divisible by 10 raised to the number of digits of n.
Worked example
Enter 376. The square is 376 × 376 = 141376 (the tool shows the square as 141376). n has three digits, so compare the last three digits of the square, 376, with n: they are identical, so the status is Automorphic. For comparison, 7 gives 49, whose last digit is 9, not 7, so 7 is not automorphic.
Common mistakes
- Comparing only the last digit. For a three-digit number such as 376 all three final digits of the square must match, not just the 6.
- Ignoring leading zeros: 625 is automorphic (625² = 390625), but a number like 0625 is the same as 625 when typed.
- Assuming that every number ending in 5 or 6 is automorphic. Only some of them are: 5 and 6, then 25, 76, 625 and 376, but 15 and 26 are not.
- Confusing automorphic numbers with palindromes or Armstrong numbers. The test here is only about the end of the square.
- Entering a number above 1,000,000 or a negative number, which gives no result.
About this calculator
An automorphic number is a number whose square ends in the number itself. For example 5² = 25 (ends in 5), 6² = 36 (ends in 6), 25² = 625 (ends in 25), and 76² = 5776 (ends in 76). This calculator squares your number and checks whether the trailing digits of the result match the original number exactly.
Automorphic numbers are a classic topic in recreational number theory: they're closely tied to modular arithmetic (a number is automorphic in base 10 if n² ≡ n mod 10^d, where d is its digit count) and to idempotent elements in the ring of integers modulo 10^d. Number theory hobbyists, puzzle-solvers and programming-challenge enthusiasts use automorphic number checks to explore patterns in digit sequences and as a classic algorithm exercise — computing large squares and comparing trailing digits efficiently.
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