More than 3
To determine how many natural numbers less than 10 divide the expression \(27^{27} - 9^{40} - 3^{79}\), we need to simplify the expression based on the prime factorization.
First, let's express each number as a power of 3:
- \(27 = 3^3\), so \(27^{27} = (3^3)^{27} = 3^{81}\).
- \(9 = 3^2\), so \(9^{40} = (3^2)^{40} = 3^{80}\).
- \(3^{79}\) remains as it is.
Substituting these back into the expression:
\(27^{27} - 9^{40} - 3^{79} = 3^{81} - 3^{80} - 3^{79}\)
This can be factored as:
\(= 3^{79}(3^2 - 3 - 1)\)
\(= 3^{79}(9 - 3 - 1)\)
\(= 3^{79} \times 5\)
Now, let's determine which natural numbers less than 10 divide \(3^{79} \times 5\). The prime factorization gives us the product of two distinct prime numbers \(3\) and \(5\):
- \(3^{79}\) indicates that the number is divisible by 3.
- \(5\) indicates divisibility by 5.
Since both prime numbers 3 and 5 divide the expression, the expression is also divisible by:
- Their product, 3 × 5 = 15, which is not less than 10, but this implies no further small prime combinations.
Four natural numbers less than 10 that divide \(3^{79} \times 5\) are:
- 1 (all numbers are divisible by 1)
- 3
- 5
- The combination of 3 and 5 is 15, include divisibility within limits, offers potential number configurations when below 10, hence individual small numbers reassert.
Thus, the correct answer is that the expression is divisible by More than 3 natural numbers less than 10.
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How many zeroes are there at the end of the following product?
1 x 5 x 10 x 15 x 20 x 25 x 30 x 35 x 40 x 45 x 50 x 55 x 60
Let XYZ be a three-digit number, where (x + y + Z) is not a multiple of 3. Then (XYZ + YZX + ZXY) is not divisible by
