Large square number

Number Theory Level 4

Find the largest possible 3-digit positive integer $N$ such that $N^2$ is of the form $ABC, ABD$ , where $A$ , $B$ , $C$ , and $D$ are (not necessarily distinct) digits and $D = C+1$ .

(An example of a number of this form is 123,124 or 998,999.)

The answer is 846.

1 solution

Patrick Corn
Nov 24, 2017

Let $M$ be the three-digit number $ABC.$ Then we want $N^2 = 1000M + M + 1 = 1001M + 1.$ So $N^2 \equiv 1$ mod $1001.$ Since $1001 = 7\cdot 11\cdot 13,$ this has eight solutions between $1$ and $1000$ by the Chinese remainder theorem , by picking various signs in the congruences $\begin{aligned} N &\equiv \pm 1 \pmod{7} \\ N &\equiv \pm 1 \pmod{11} \\ N &\equiv \pm 1 \pmod{13} \end{aligned}$ The eight solutions are $N = 1,1000; 428, 573; 274, 727; 155, 846.$ So the largest three-digit one is $\fbox{846}.$

Large square number

The answer is 846.

1 solution

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