Find the smallest positive integer n 10 such that n+6 is a prime and 9 n+7 is a perfect square.

Step-by-step solution

Since 9 n+7 is a perfect square Let us assume that 9 n+7= m ^2 Also n+6 is a prime number n+6 must be odd number n also must be an odd number So Let us assume n =2 k +1 ....(ii) where k is an Integer aligned & 9(2 k+1)+7=m^2 & 18 k=m^2-16 & 18 k=(m+4)(m-4) ...(iii) aligned Since 18 k is even and m+4 and m-4 both are of some parity m must also be even. say m=2 p , where p is an Integer Substituting in (iii) aligned & 18 k=(2 p+4)(2 p-4)=4(p+2)(p-2) & 9 k=2(p+2)(p-2) aligned k must be even, say k =2 where is an Integ