Real Numbers

### Show that every positive even integer is of the form (6m+1) or (6m+3) or (6m+5)where m is some integer.

Solution: Let's say that $n$ be any arbitrary positive odd integer. Upon dividing $n$ by 6, let $m$ be the quotient and $r$ be the remainder. Therefore, by Euclid's division lemma, we get $n=6 m+r$,...

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### Using Euclid’s algorithm, find the HCF of (i) 405 and 2520 (ii) 504 and 1188

Solution: (i) When applying Euclid's algorithm, that is dividing $2520$ by $405$, we obtain, Quotient $=6$, Remainder $=90$ $\therefore 2520=405 \times 6+90$ Again upon applying Euclid's algorithm,...

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### What do you mean by Euclid’s division algorithm?

Solution: It is stated by the Euclid's division algorithm that for any two positive integers $a$ and $b$, there exist unique integers $q$ and $r$, such that $a=b q+r$. where $0 \leq r \leq b$.

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### The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p q what can you say about the prime factors of q? (i) 43.123456789 (ii) 0.120120012000120000. . .(iii)

(i) 43.123456789 Since it has a terminating decimal expansion, it is a rational number in the form of p/q and q has factors of 2 and 5 only. (ii) 0.120120012000120000. . . Since, it has...

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### Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.

Solution: According to Euclid’s division Lemma, Let n be the positive integer And b equals to 3 Where q is the quotient and r is the remainder $n\text{ }=3q+r$, 0<r<3 implies remainders may be...

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### Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.

Solution: Where q is an integer and r = 0, 1, 2, 3, 4, 5, then 6q + r is a positive integer. The positive integers are then of the form: 6q, 6q+1, 6q+2, 6q+3, 6q+4, and 6q+5. Taking cube on L.H.S...

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### Show that the square of any odd integer is of the form 4q + 1, for some integer q.

Solution: Let b=4 and a be any odd integer. According to Euclid’s algorithm, For some integer $m\ge ~0,\text{ }a=4m+r$ And r = 0,1,2,3 as 0 ≤ r < 4. As a result we have, a = 4m or 4m +...

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### 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, Number of army contingent members=616 Number of army band members = 32 When we find the HCF(616,32) then we get  maximum number of columns in which they can march. We use Euclid’s...

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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.

Let a be any positive integer and b=6.Then,by Euclid’s algorithm, $a=6q+r$ for some integer $q\ge 0$ and r=0,1,2,3,4,5 because $0\le r<6$ . Therefore, $a=6q$ or $6q+1$ or  $6q+2$ or $6q+3$...

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### Use Euclid’s division algorithm to find the HCF of:

(1)We have 225>135 So, we apply the division lemma to 225 & 135 to obtain 225=135×1+90 Here remainder 90≠0, we apply the division lemma again to 135 and 90 to obtain 135=90×1+45 We consider...

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