What do you mean by Euclid’s division lemma?
A number, when divided by 61, gives 27 as quotient and 32 as remainder. Find the
number.
By what number should 1365 be divided to get 31 as quotient and 32 as
remainder?
Using Euclid’s algorithm, find the HCF of:
(i) 405 and 2520 (ii) 504 and 1188 (iii) 960 and 1575
Show that every positive integer is either even or odd.
Show that any positive odd integer is of the form (6m + 1) or (6m + 3) or (6m
Show that any positive odd integer is of the form (4m + 1) or (4m + 3), where
m is some integer.
For any positive integer n, prove that n^3 – n is divisible by 6.
Prove that if x and y are both odd positive integers then x^2 + y^2 is even
but not divisible by 4.
Use Euclid’s algorithm to find HCF of 1190 and 1145. Express the HCF in
the form 1190m + 1445n.
Question 1: Using prime factorization, find the HCF and LCM of:
(i) 36, 84 (ii) 23, 31 (iii) 96, 404
(iv) 144,198 (v) 396, 1080 (vi) 1152, 1664
In each case, verify that:
HCF x LCM = Product of given numbers
Using prime factorization, find the HCF and LCM of:
(i) 8, 9, 25
(ii) 12, 15, 21
(iii) 17, 23, 29
(iv) 24, 36, 40
(v) 30, 72, 432
(vi) 21, 28, 36, 45
The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find
the other.
The HCF of two numbers is 145 and their LCM is 2175. If one of the numbers is 725, find the other.
The HCF of two numbers is 18 and their product is 12960. Find their LCM.
Is it possible to have two numbers whose HCF is 18 and LCM is 760? Give
reason.
Find the simplest form of:
(i) 69/92
(ii) 473/645
(iii) 1095/1168
(iv) 368/496
Find the largest number which divides 438 and 606, leaving the remainder 6 in each case.
Find the largest number which divides 320 and 457, leaving remainders 5 and 7
respectively.
Find the least number which when divided by 35, 56, and 91 leaves the
same remainder 7 in each case.
Find the smallest number which when divided by 28 and 32 leaves
remainders 8 and 12 respectively.
Find the smallest number which when increased by 17 is exactly divisible
by both 468 and 520.
Find the greatest number of four digits which is exactly divisible by 15, 24
and 36.
Find the largest four-digit number which when divided by 4, 7 and 13
leaves a remainder of 3 in each case.
Find the least number which should be added to 2497 so that the sum is
exactly divisible by 5, 6, 4, and 3?
Find the greatest number that will divide 43, 91, and 183 so as to leave the
the same remainder in each case.
Find the greatest number which when divided by 20, 25, 35, and 40 leaves
remainder as 14, 19, 29, and 34 respectively.
In a seminar, the number of participants in Hindi, English, and mathematics
are 60, 84, and 108 respectively. Find the minimum 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.
Three sets of English, Mathematics and Science books containing 336, 240
and 96 books respectively have to be stacked in such a way that all the books are
stored subjectwise and the height of each stack is the same. How many stacks will be
there?
Three pieces of timber 42 m, 49 m, and 63 m long have to be divided into planks of the
same length. What is the greatest possible length of each plank? How many planks are
formed?
Find the greatest possible length which can be used to measure exactly
the lengths 7 m, 3 m 85 cm, and 12 m 95 cm.
Find the maximum number of students among whom 1001 pens and 910
pencils can be distributed in such a way that each student gets the same number of
pens and the same number of pencils.
Find the least number of square tiles required to pave the ceiling of a room
15 m 17 cm long and 9 m 2 cm broad.
Three measuring rods is 64 cm, 80 cm, and 96 cm in length. Find the least
length of cloth that can be measured an exact number of times, using any of the rods.
An electronic device makes a beep every 60 seconds. Another device
makes a beep after every 62 seconds. They beeped together at 10 a.m. At what time will
they beep together at the earliest?
The traffic lights at three different road crossings change after every 48
seconds, 72 seconds, and 108 seconds respectively. If they all change simultaneously at
8 a.m., then at what time will they again change simultaneously?
Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12
minutes respectively. In 30 hours, how many times do they toll together?
Without actual division, show that each of the following rational numbers is a
terminating decimal. Express each in decimal form.
Without actual division, show that each of the following rational numbers is
a non-terminating repeating decimal:
Express each of the following as a fraction in simplest form:.
Define (i) rational numbers, (ii) irrational numbers, (iii) real numbers.
Classify the following numbers as rational or irrational:
Prove that each of the following numbers is irrational.
Prove that 1/â?š3 is irrational.
(i) Give an example of two irrationals whose sum is rational.
(ii) Give an example of two irrationals whose product is rational.
State whether the given statement is true or false.
(i) The sum of two rationals is always rational.
(ii) The product of two rationals is always rational.
(iii) The sum of two irrationals is always irrational.
(iv) The product of two irrationals is always irrational.
(v) The sum of a rational and irrational is irrational.
(vi) The product of rational and irrational is irrational.
Prove that (2�3 – 1) is an irrational number.
Prove that (4 – 5â?š2 ) is an irrational number.
Prove that (5 – 2�3) is an irrational number.
Prove that 5â?š2 is irrational.
Prove that 2/â?š7 is irrational.
State Euclid’s division lemma.
State fundamental theorem of Arithmetic.
Express 360 as a product of its prime factors.
If a and b are two prime numbers, then find HCF (a, b).
If a and b are two prime numbers then find LCM (a, b).
If the product of two numbers is 1050 and their HCF is 25, find their LCM.
What is a composite number?
If a and b are relatively prime then what is their HCF?
If the rational number a/b has a terminating b decimal expansion, what is the condition to be satisfied by b.
Simplify
Write the decimal expansion of
Show that there is no value of n for which (2n x 5n ) ends in 5.
Is it possible to have two numbers whose HCF is 25 and LCM is 520?
Give an example of two irrationals whose sum is rational.
Give an example of two irrationals whose product is rational.
If a and b are relatively prime, what is their LCM?
The LCM of two numbers is 1200. Show that the HCF of these numbers cannot be 500. Why?
Express why 0.15015001500015… is an irrational number.
Show that â?š2/3 is irrational.
Write a rational number between â?š3 and 2.
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