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Math Prime number puzzle 15april

math puzzle
Puzzle

If 'a' is a prime number, then prove, that a*a+26 is NOT a prime number (whether it is true).

For Solution: Click Here

10 comments:

  1. 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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  3. a*a + 26 = (a-2)(a+2) + 30

    we can directly check for a equal to 3,

    a*a+26 = 35 [NOT PRIME]

    for consecutive 5 numbers scenario, consider them as below

    a-2, a-1, a, a+1, a+2
    O1, E1, O, E2, O2

    here O=Odd, E=even

    out of O1, E1, and O
    at least one is divisible by 3

    if O1 is divisible by 3 then (a-2)(a+2) is divisible by 3
    if E1 is divisible by 3 then O2 is divisible by 3 and hence (a-2)(a+2) is divisible by 3

    so 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

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  4. True.
    n=3:
    3*3+26=35=5*7
    n=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)

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  5. 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.

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  6. 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*6
    But A num 'a' is said to be a primenumber if it has only two multiples i.e., 1 and a only

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  7. If A is even then A^2+26 is even
    If 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.

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