Euclid's Division Lemma

Q.

144 cartons of Coke Cans and 90 cartons of Pepsi Cans are to be stacked in a Canteen. If each stack is of the same height and is to contain cartons of the same drink, what would be the greatest number of cartons each stack would have?

Q.

Use Euclids division lemma to show that the square of any positive integer is either of the form $3mor3m+1$ for some integer $m$.

Q. Prove that the product of three consecutive positive integers is divisible by 6

Q.

Find the HCF of $65$and $117$ and express it in form of $65m+117n$

Q.

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

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.

Find the HCF of the following pairs of integers and express it as a linear combination of them.
(i) 963 and 657 (ii) 592 and 252 (iii) 506 and 1155
(iv) 1288 and 575

Q.

Define HCF of two positive integers and find the HCF of the following pairs of numbers.
(i) 32 and 54 (ii) 18 and 24 (iii) 70 and 30
(iv) 56 and 88 (v) 475 and 495 (vi) 75 and 243
(vii) 240 and 6552 (viii) 155 and 1385 (ix) 100 and 190
(x) 105 and 120

Q.

Prove that (n³- n) is divisible by 6

Q.

A number when divided by 143 leaves 31 as remainder. What will be the remainder when the same number is divided by 13 ?

(a) 0 (b) 1 (c) 3 (d) 5

Q.

What is the HCF of two consecutive numbers?

Q.

Show that 1.272727...=1.27 can be expressed in the form of p/q, where p and q are integers and q≠0

Q.

The largest 4 digit number exactly divisible by 88 is:

1. 9944

2. 9988

3. 9966

4. 8888

Q.

Write whether every positive integer can be of the form $4q+2$, where $q$ is an integer. Justify your answer.

Q.

If the H.C.F. of $65$ and $117$ is expressible in form $65m–117$, calculate the value of $m$?

Q.

The numbers $11284$ and $7655$ when divided by a certain number of three digits, leave the same remainder. find that three digit number.

Q.

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

1. False

2. True

Q. In the Euclid's Division Lemma, if divident is 42, divisor is 5 and remainder is 2 times of its quotient then, find the quotient.

Q.

What does $R$ mean in math?

Q. If x is natural number which when divided by 5, leaves remainder 3, then the remainder when $x^2-5x+2$ is divided by 5 is

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

Q.

Find $q$ and $r$ for the following pairs of positive integers $a$ and $b$ satisfying $a=bq+r$.

(1) $a=13,b=3$

(2) $a=8,b=80$

(3) $a=125,b=5$

(4) $a=132,b=11$.

Q.

How many three digit numbers have exactly three factors?

Q. Use Euclid’s algorithm to find the H.C.F. of 420 and 130.

1. 11
2. 12
3. 13
4. 10

Q.

The square of one number is $25$. If the HCF and LCM of the two numbers are $5$ and $35$ find another number.

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. True

2. False

Q.

If the HCF of 408 and 1032 is expressible in the form 1032 $m-408\times 5,$ find m.

Q.

Find that sum of first $30$ integer divisible by $6$.

Q.

There are$527\mathrm{apples},646\mathrm{pears},\mathrm{and}748\mathrm{oranges}$. These are to be put in heaps of equal quantities. Find the maximum number of fruits in each heap. How many such heaps would be formed

Q. If p/q is a rational number (q is not equal to 0), what is the condition on q, so that the decimal representation of p/q is terminating?

Q.

In Euclids Division Lemma when $a=bq+r$ where $a,b$ are positive integers then what values $r$ can take?