Let p be a prime number greater than 3. Which of the following expressions is always divisible by 24 and necessarily by 48?
\(p^4 - 1\)
Every prime p greater than 3 is odd and not a multiple of 3. So (p – 1) and (p + 1) are consecutive even numbers (one of them a multiple of 4), and one of them is a multiple of 3; therefore \(p^2 - 1 = (p - 1)(p + 1)\) is always divisible by 8 × 3 = 24.
But \(p^2 - 1\) need not be divisible by 48: for p = 5 it equals 24. Also \(p^2 + 1\) (26 for p = 5) and \(p^2 - p\) (20 for p = 5) are not even divisible by 24.
Now \(p^4 - 1 = (p^2 - 1)(p^2 + 1)\). The first factor is a multiple of 24 and the second factor \(p^2 + 1\) is even (p is odd), so the product is a multiple of 24 × 2 = 48.
Therefore only \(p^4 - 1\) is always divisible by both 24 and 48.
Check: p = 5 gives 625 – 1 = 624 = 48 × 13, and p = 7 gives 2401 – 1 = 2400 = 48 × 50; both are multiples of 24 and of 48.
Hence, \(p^4 - 1\).
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Which of the following is/are correct?
1. S is always divisible by 74.
2. S is always divisible by 9.
select the correct answer using the code given below:
How many 3-digit natural numbers (without repetition of digits) are there such that each digit is odd and the number is divisible by 5?
