Euclid's Division Lemma

Q. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
[3 marks]
[NCERT]
[Euclids division Algoritham]

Q.

A number when divided by 6 leaves a remainder 3. When the square of the number is divided by 6, the remainder is 3.

1. 3

2. 1

Q.

According to Euclid's division lemma, if $a$ and $b$ are two positive integers with $a>b,$ then which of the following is true? (Here, $q$ and $r$ are unique integers.)

1. $a=b\times r$ where $0\le r<b$

2. $b=aq+r$ where $0\le r<a$

3. $a=bq+r$ where $0\le r<b$

4. $a=b+r$ where $0\le r<a$

Q.

Every positive odd integer is of the form 2q+1, where q is some integer.

1. False

2. True

Q.

What is the least number that must be added to 1056 so the number is divisible by 23?

1. 2
2. 3
3. 0
4. 1

Q.

The product of $3$ consecutive numbers, when the first number is even is divisible by ___

1. all of these

2. $24$

3. $12$

4. $6$

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.

If $p=a^x.b^y.c^z$ and $q=3^x.b^4{.5}^z$ where a, b, c are primes and x, y, z are natural numbers, then which of the following is true if p=q ?

1. a + c = 2y

2. a = 3x

3. a = b = c

4. x = y = z

Q.

If $2^p{.3}^q{.5}^r=60$, which of the following is true? Given; p, q, r are natural numbers.

1. p + q + r = 0

2. p = q = r

3. p = 2 , q = r = 1

4. p + q + r = 3

Q.

$\text{If the number}{\mathrm{1.2.2}}^2{.2}^3{.2}^4..........{.2}^{20}.{3.3}^2{.3}^3{.3}^4.......{.3}^{10}$
$\text{is written in power of '6' then the highest}$
$\text{power of '6' is \_\_\_\_\_}.$

1. 10

2. 20

3. 55

4. 30