Question

Show that 101 is a factor of 87³ + 14³.

Answer 1

We are asked to show that 101 is a factor of 87³ + 14³.

Step 1: Apply the sum of cubes formula

We can use the sum of cubes formula:

a³ + b³ = (a + b)(a² – ab + b²)

In this case, let a = 87 and b = 14.

Applying the formula:

87³ + 14³ = (87 + 14)(87² – 87 × 14 + 14²)

Step 2: Simplify the first part

First, simplify 87 + 14:

87 + 14 = 101

Now, we know that the expression is:

87³ + 14³ = 101 × (87² – 87 × 14 + 14²)

Step 3: Simplify the second part

Next, calculate the terms inside the parentheses:

87² = 7569,

87 × 14 = 1218,

14² = 196

Now, substitute these values into the expression:

87² – 87 × 14 + 14² = 7569 – 1218 + 196

Simplify:

7569 – 1218 + 196 = 6547

Step 4: Final expression

Now, we have:

87³ + 14³ = 101 × 6547

Step 5: Conclusion

Since 87³ + 14³ = 101 × 6547, we have shown that 101 is indeed a factor of 87³ + 14³.

Thus, 101 is a factor of 87³ + 14³.