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Using the Euclidean division lemma, what is the highest common factor (HCF) of the numbers 625, 3125, and 15625?

Answer :

Final answer:

The highest common factor (HCF) of numbers 625, 3125, and 15625 is 625. This is determined by their prime factorizations. No need of using the Euclidean division lemma.

Explanation:

The question is asking to find the Highest Common Factor (HCF) of numbers 625, 3125, and 15625 using the Euclidean division lemma. The Euclidean division lemma is a theorem on elementary number theory which states that given two positive integers a and b, there exist unique integers q and r such that a = bq + r, 0 ≤ r < b. The HCF of two numbers can be determined using this theorem. But for these particular numbers 625, 3125, and 15625, you don't necessarily need the Euclidean division lemma to find their HCF, you can use the basic definition of HCF. The definition says the highest number that divides all the given numbers is their HCF.

The prime factorization of 625 is 5^4, of 3125 is 5^5, and of 15625 is 5^6. The highest number that can divide all three numbers is the smallest power of 5, that is 5^4 = 625. So, the HCF of 625, 3125, 15625 is 625.

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