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Real Numbers
An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
Given integers are 32 and 616.
Clearly 616 > 32. Therefore, applying Euclid’s division lemma to 616 and 32, we get
Since, the remainder 8 ≠ 0, we apply the division lemma, to get
The remainder has now become zero, so our procedure stops. Since the divisor at this stage is 8.
Therefore, the maximum number of columns in which both 616 members (army contingent) and 32 members (army band) can march is 8.
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Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Express each number as a product of its prime factors: (i) 140
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Express each number as a product of its prime factors: (v) 7429
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