Suppose that $a$ is a positive integer for which the least common multiple of $a+1$ and $a-5$ is $10508$. What is $a^2 - 4a + 1$?

21022.

Step-by-step explanation:

Find the prime factors of 10508:

2) 10508

2) 5254

37) 2627

71.

50208 = 2*2*37*71.

Now there is no integer value for a that would fit (a + 1) (a - 5) = 10508.

But we could try multiplying the LCM by 2:-

= 21016 = 2*2*2*37*71.

= 2*2*37 multiplied by 2 * 71

= 148 * 142.

That looks promising!

a - 5 = 142 and

a + 1 = 148

This gives 2a - 4 = 290

2a = 294

a = 147.

So substituting a = 147 into a^2 - 4a + 1 we get:

= 21022.