555

Not sure what you want with “555.” Here are a few quick notes about the number, plus some ready-to-go practice problems you can use for exam prep. Tell me your subject area (math, algebra, number theory, etc.) and I’ll tailor more.

Quick facts about 555

• Prime factorization: 555 = 3 × 5 × 37
• Divisors: 1, 3, 5, 15, 37, 111, 185, 555
• Sum of divisors σ(555) = (1+3)(1+5)(1+37) = 4 × 6 × 38 = 912
• Euler’s totient: φ(555) = 555 × (1 − 1/3) × (1 − 1/5) × (1 − 1/37) = 288
• Digits sum = 5 + 5 + 5 = 15, so divisible by 3 and by 5

Sample problems (with solutions)

1. Find all divisors of 555.

• Answer: 1, 3, 5, 15, 37, 111, 185, 555

2. Solve the quadratic equation x^2 + x − 555 = 0.

• Solution: Discriminant D = 1 + 4·555 = 1 + 2220 = 2221 x = [−1 ± sqrt(2221)]/2 (Note: 2221 is not a perfect square, so the roots are irrational.)

3. Compute the number of integers between 1 and 555 that are coprime to 555.

• Answer: φ(555) = 288

4. If integers a and b satisfy a + b = 555, what is the maximum possible value of ab?

• To maximize ab with fixed sum, take a and b as close as possible: a = 277, b = 278
• Maximum product: 277 × 278 = 77,006

5. Find the sum of all divisors of 555.

• Answer: σ(555) = 912

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