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True! If 'a' is a prime number then a*a+26 will not be a prime number.Proof: if 'a' is not a prime number then it would not be divisible by 3. After dividing by 3 it gives remainder '1' or '2' then a*a % 3 would be 1 and 1+26=27 that is divisible by 3. so "a*a+26" is not a prime number bcoz it is divisible by 3.
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a*a + 26 = (a-2)(a+2) + 30we can directly check for a equal to 3,a*a+26 = 35 [NOT PRIME]for consecutive 5 numbers scenario, consider them as belowa-2, a-1, a, a+1, a+2O1, E1, O, E2, O2here O=Odd, E=evenout of O1, E1, and Oat least one is divisible by 3if O1 is divisible by 3 then (a-2)(a+2) is divisible by 3if E1 is divisible by 3 then O2 is divisible by 3 and hence (a-2)(a+2) is divisible by 3so we can write the equation (a-2)(a+2) + 30 as=> 3K + 30=> 3(K+10)thus the number is divisible by 3 if a is not equal to 3, and hence is NOT PRIME
True.n=3: 3*3+26=35=5*7n=3k+1: (3k+1)*(3k+1)+26=3*(k(3k+2)+9)n=3k-1: (3k-1)*(3k-1)+26=3*(k*(3k-2)+9)
Let us divide it by 3. Its a prime number, So dividing it by 3 would result in 1 or 2 as remainder. If 1 is remainder and 26 is added to it. result would be 27 and it would be divisible by 3 So not a prime number. If 2 is remainder and 26 is added to it. result is 28 which is clearly not a prime number. So a * a + 26 is not a prime number.
It is true.for that consider a=2(primenumber)substitute the value of a in a*a+26 then 2*2+26=4+26=30 (WHICH IS NOT A PRIME)Because 30=1*30 2*15 3*10 5*6But A num 'a' is said to be a primenumber if it has only two multiples i.e., 1 and a only
If A is even then A^2+26 is evenIf A is NOT a multiple of 3 then A^2 is 1 Mod 3 so A^2+26 is a multiple of 3. This leaves only odd multiples of 3 as candidates of which only 3 is prime. 3^2+26=35 which is not prime.