Show that any positive odd integer is of the form (6m + 1) or (6m + 3) or (6m
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
Find the least number which when divided by 35, 56, and 91 leaves the
same remainder 7 in each case.
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
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
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
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?
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:
(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.
If the rational number a/b has a terminating b decimal expansion, what is the condition to be satisfied by b.
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