2015 · Q39
Consider the following statements :
- 1There exists only one prime number p such that (17p + 1) is a square. 2. If x is the product of 10 consecutive prime numbers starting from 2, then (x+ 1) is also a prime number.
Which of the above statements is/are correct?
Elementary MathematicsNumber System
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Statement 1: 17p + 1 = k² gives (k−1)(k+1) = 17p; the factor pairs of 17p differing by 2 admit only k − 1 = 17, k + 1 = p, i.e. k = 18, p = 19 (check: 17×19+1 = 324 = 18²) — true, with p = 19 unique. Statement 2: 2·3·5·7·11·13·17·19·23·29 + 1 = 6469693231 is composite (verified computationally), so it is not prime — statement 2 is false. Hence '1 only' (option A). Stored key corrected from C to A.