Real Numbers
Questions and Answers
Questions and Answers
Q. 1. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
Sol.
By Euclid’s algorithm,
a = 6q + r, and r = 0, 1, 2, 3, 4, 5
Hence, a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5
`Clearly, 6q + 1, 6q + 3, 6q + 5 are of the form 2k + 1, where k is an integer.
Therefore, 6q + 1, 6q + 3, 6q + 5 are not exactly divisible by 2.
Hence, these numbers are odd numbers.
Sol.
By Euclid’s algorithm,
a = 6q + r, and r = 0, 1, 2, 3, 4, 5
Hence, a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5
`Clearly, 6q + 1, 6q + 3, 6q + 5 are of the form 2k + 1, where k is an integer.
Therefore, 6q + 1, 6q + 3, 6q + 5 are not exactly divisible by 2.
Hence, these numbers are odd numbers.
Q. 2. 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?
Sol.
Euclid’s algorithm
616 = 32 × 19 + 8
32 = 8 × 4 + 0
The HCF (616, 32) is 8.
Therefore, they can march in 8 columns each.
Euclid’s algorithm
616 = 32 × 19 + 8
32 = 8 × 4 + 0
The HCF (616, 32) is 8.
Therefore, they can march in 8 columns each.
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