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19 December, 00:33

Prove the divisibility of the following numbers:

25^9 + 5^7 is divisible by 30.

Also, read as (25 to the power of 9) + (5 to the power of 7) is divisible by 30.

Answer: Blank x 30

What is the blank? (It should be expressed in exponent form)

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Answers (1)
  1. 19 December, 03:33
    0
    Step-by-step explanation:

    prove : 25^9 + 5^7 ≡ 0 (mod 30) or 5^18 + 5^7 ≡ 0 (mod 30)

    because : 25 = 5²

    calculate 5^p for 1; 2; 3; 4 ...

    5^1 ≡ 5 (mod 30)

    5^2 ≡ 25 (mod 30)

    5^3 ≡ 5 (mod 30)

    5^4 ≡ 25 (mod 30)

    p = 2k+1 5^p ≡ 5 (mod 30)

    p = 2k 5^p ≡ 25 (mod 30)

    so : 5^18 ≡ 25 (mod 30) ... (*)

    5^7 ≡ 5 (mod 30) ... (**)

    add (*) and (**) : 5^18 + 5^7 ≡ 0 (mod 30) because : 30≡0 (mod30)
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