N is an even 3-digit number such that the last 3 digits of N*N are N itself. What is the value of N?
To find the value of N, we need to first understand the requirements given in the question. Here are the steps to solve the problem:
1. Let's assume the 3-digit number N has the form ABC, where A, B, and C are its digits. For example, if N is 123, A = 1, B = 2, and C = 3.
2. Since N is an even number, C must be an even digit (0, 2, 4, 6, or 8). Otherwise, N would be an odd number.
3. We know that N multiplied by itself is equal to N. Mathematically, this can be written as N * N = N.
4. Expanding this equation using the assumption from step 1, we get (100A + 10B + C) * (100A + 10B + C) = 100A + 10B + C.
5. Multiplying the left side of the equation, we have (10000A^2 + 2000AB + 200AC + 1000B^2 + 100BC + C^2) = 100A + 10B + C.
6. By comparing the coefficients of the powers of 10, we can deduce the following equations:
- 10000A^2 = 100A (because the coefficient of A^2 term is 10000 and the coefficient of A term is 100)
- 2000AB + 200AC = 10B (because the coefficient of AB term is 2000 and the coefficient of B term is 10)
- 1000B^2 + 100BC = C (because the coefficient of B^2 term is 1000 and the coefficient of C term is 1)
- C^2 = C (since there is no C term on the left side)
7. From the fourth equation, we know that C^2 = C. Solving this equation gives two possibilities for C: C = 0 or C = 1.
8. If C = 0, then the third equation becomes 1000B^2 = 0, which means B = 0.
9. Now, substituting C = 1 into the third equation gives 1000B^2 + 100B = 1. Solving this equation for B, we find B = 0 or B = 1.
10. For B = 0, the second equation becomes 2000A = 0, which means A = 0. However, we assumed A, B, and C to be digits, so A cannot be 0.
11. Therefore, for B = 1, the second equation gives 2000A + 200A = 10. Solving this equation for A, we find A = 1/220 = 5.
12. From the results of step 11, we obtain N = ABC = 510.
Therefore, the value of N is 510.