Euclid's Division Lemma

Q.

Write whether the square of any positive integer can be of the form $3m+2$, where $\mathrm{m}$ is a natural number. Justify your answer.

Q.

Express the following in the form, where p and q are integers and q ≠ 0.

(i) (ii) (iii)

Q.

Two numbers a and b, when divided by 7 and 6 respectively, leave remainders p and q respectively. What is the maximum value of p + q ?

1. 11

2. 12

3. 6

4. 5

Q.

What do you mean by Euclids division lemma?

Q. Show that every positive integer is of the form $2q$, and that every positive odd integer is of the form $2q+1$, where $q$ is some integer.

Q. For any integers $a$ and $3$, there exists unique integers $q$ and $r$ such that $a=3q+r$. Find the possible value of $r$.

Q.

Express the following as the sum of two odd primes: $44$

Q. According to Euclid's division Lemma for the two polynomials
$s\left(x\right)$ and $r\left(x\right)$ , we can write as $s\left(x\right)=p\left(x\right)\times q\left(x\right)+r\left(x\right)$. which of the following about $r\left(x\right)$ is/are true ?

1. $r\left(x\right)$ can be zero
2. Degree of $r\left(x\right)$ <0
3. Degree of $r\left(x\right)$ >0
4. $r\left(x\right)$ cannot be zero

Q.

For an odd positive integer to be perfect square it should be necessarily be of the 8k + 1 unable to understand this line

Q. What is the largest number that divides 245 and 1029, leaving remainder 5 in each case ?

1. 16

Q. $\pi$ is a ___ number.

1. Rational
2. Irrational
3. Integer
4. None of these

Q.

6-4 root 3 / 6+4 root 3

Q. Rational number from following real numbers $-8.0,\sqrt{5},\frac{5}{7},-\sqrt{18},\sqrt{32},4.28,\pi ,3,\frac{8}{15},0.075$ are $-\mathrm{8.0.}\frac{5}{7},4.28,3,\frac{8}{15},0.075$.

1. True
2. False

Q. If $x^{51}+51$ is divided by $x+1$, the remainder is

Q. The relation of Euclid’s division lemma is

1. $a=bq+r$
2. $a=b+q\times r$
3. $a=bqr$
4. $a=(b–q)r$

Q. Root 7-1 divided by root 7+1 - root 7+1 divide by root 7-1

Q. If $a=107,b=13$, using Euclid's division algorithm find the values of ${}^{\prime }{q}^{\prime }$ and ${}^{\prime }{r}^{\prime }$ such that $a=bq+r$.

Q. State "Euclid's " division lemma?

Q. 65 and 495

Q. According to Euclid's division lemma, for given positive integers $a$ and $7$, there exit unique non negative integers $q$ and $r$ such that $a=7q+r$, where__________

1. $0<r\le 7$
2. $1\le r\le 7$
3. $0\le r<7$
4. $r\ge 7$

Q. Using Euclid's division algorithm find the HCF of the following numbers.
$2024$ and $1872$

Q.

Express the following as the sum of two odd primes: $18$

Q. In Euclid's division algorithm, $a=bq+r(a,b,q,r\in Z,b\ne 0),r$ satisfies the condition____

1. $0\le r<b.$
2. $0\le r<|b|$
3. $0\le r\le |b|$
4. $0<r\le b$