Identify a single-digit number other than 1, which may exactly divide the number 17³ + 18³ − 16³ − 15³.

Given:

Identify a single-digit number other than 1, which may exactly divide the number 17³ + 18³ - 16³ - 15³.

Formula used:

We will use the fact that the expression (a³ + b³ - c³ - d³) may be simplified by grouping and factoring.

Calculation:

Let A = 17, B = 18, C = 16, D = 15

Expression: A³ + B³ - C³ - D³

We use the identity for sum and difference of cubes:

⇒ (A³ + B³) - (C³ + D³)

We further simplify using the factorization:

⇒ (A + B)(A² - AB + B²) - (C + D)(C² - CD + D²)

Substitute the values:

⇒ (17 + 18)((17²) - (17×18) + 18²) - (16 + 15)((16²) - (16×15) + 15²)

⇒ 35(289 - 306 + 324) - 31(256 - 240 + 225)

⇒ 35(307) - 31(241)

⇒ 10745 - 7471

⇒ 3274

Now check the divisibility of 3274 by the given options:

3274 ÷ 2 = 1637 (exact division)

3274 ÷ 3 = 1091.33 (not exact)

3274 ÷ 7 = 467.71 (not exact)

3274 ÷ 5 = 654.8 (not exact)

∴ The correct answer is option 1.