Euclid's Division Algorithm

Q.

A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number of sweets, and they take up the least area of the tray. What is the maximum number of barfis that can be placed in each stack for this purpose?

1. 6

2. 13

3. 10

4. 7

Q. If two positive integers m and n are expressible in the form m = pq³ and n = p³q², where p, q are prime numbers, then HCF (m, n) =
(a) pq
(b) pq2
(c) p³q²
(d) p²q²

Q.

Use Euclids division algorithm to find the HCF of $4052$ and $12576$.

Q.

Using euclids division algorithm find whether 847 and 2160 are co prime or not.

Q.

Prove that if x and y are both odd positive integers then X square + Y square is even but not divisible by 4

Q.

A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

Q. Use Euclids division algorithm to find the HCF of
(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 255
(iv) 184, 230 and 276
(v) 136, 170 and 255

Q.

A number when divided by a divisor leaves a remainder of $24$, when twice of the original number is divided by the same divisor the remainder is $11$, then find the divisor.

1. $13$

2. $59$

3. $35$

4. $37$

Q. Apply division algorithm to find the quotient q(x) and remainder r(x) in dividing f(x) by g(x) in each of the following :

(i) f(x) = x³ − 6x² + 11x − 6, g(x) = x² + x + 1
(ii) f(x) = 10x⁴ + 17x³ − 62x² + 30x − 3, g(x) = 2x² + 7x + 1
(iii) f(x) = 4x³ + 8x + 8x² + 7, g(x) = 2x² − x + 1
(iv) f(x) = 15x³ − 20x² + 13x − 12, g(x) = 2 − 2x + x²

Q.

Renu purchases two bags of fertilizer of weights$75$ kg and$69$ kg. Find the maximum value of weight which can measure the weight of the fertilizer an exact number of times.

Q.

Show that only one out of n, n+3, n+6, n+9 is divisible by four

Q.

Find two consecutive odd positive integers, the sum of whose squares is $290$ by using the quadratic formula.

Q.

Question 3
Show how $\sqrt{5}$ can be represented on the number line.

Q. If 'd' is the HCF of 30, 72, find the value of 'x' and 'y' satisfying d=30x+72y.

Q.

Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.

Q. Find the largest number that will divide $398,436$ and $542$ leaving remainders $7,11$ and $15$ respectively.

Q.

Find three consecutive odd integers , the sum of whose squares is 83

Q. In a seminar, the number of participants in Hindi, English and Mathematics are $60,84$ and $108$ respectively. Find the number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.

Q.

If $d$ is $\mathrm{HCF}$ of $56$ and $72$, then find $x,y$ satisfying the equation $d=56x+72y$. Also show that $x$ and $y$ are not unique.

Q.

Prove that X² -X is divisible by 2for all positive integer X

Q. Question 4
If the HCF of 65 and 117 is expressible in the form 65m - 117, then the value of m is

A) 4
B) 2
C) 1
D) 3

Q.

Prove that the square of any positive integer of the form 5q + 1 is of the same form.

Q. If d is the HCF of 160 and 24, then find x and y satisfying $d=160x+24y$

1. 23 and -153
2. 43 and -153
3. 33 and 153
4. -153 and 23

Q.

Show that one and only one out of n, n+1, n+4 is divisible by 3 where n is any positive integer.

Q.

Question

Prove that n^2-n is divisible by 2 for every positive integer n

Q.

Divide $3605$ by$29$and verify the division algorithm

Q. If the $HCF$ of $210$ and $55$ is expressible in the form $210\times 5+55y,$ find $y.$

Q.

What is the square root of $1600$?

Q.

Find the greatest number that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.

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