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Show that any positive odd integer is of form 4m + 1 or 4m + 3, where m is some integer.
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Let positive integer be
If
This is an odd integer
If
This is also an odd integer.
Show that any positive odd integer is of the form 6m + 1, or 6m + 3, or 6m + 5, where m is some integer.
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If
It is always an odd integer.
If
It is also an odd integer
If
It is also an odd integer.
Find the LCM of 2520 and 2268 by prime factorisation.
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LCM
Show that \(5\sqrt3 \) in irrational
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Let us assume that is rational number.
and are integer so is rational but is irrational. This contradicts the assumption. So is irrational number.
Write the denominator of the rational number \(\dfrac{257}{5000} \) in the form \(2^m\times5^n \), where m, n are nonnegative integers. Hence, write its decimal expansion, without actual division.
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Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times.
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Maximum capacity = HCF of 850 and 680
HCF
Without actually performing the long division, find if \(\dfrac{987}{10500} \) will have terminating or nonterminating repeating decimal expansion. Give reason for your answer.
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Yes,
Explain why \(7\times11\times13+13 \) and \(7\times6\times5\times4\times3\times2\times1+5 \) are composite numbers.
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Can the number \( 6^n,\ n\) being a natural number, end with the digit \(5 \)? Give reasons.
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For any number ending with 5 the prime factor 5 should be there in its prime factorisation. But has only and as prime factors. So can never end with digit
In the given factor tree, find the numbers m, n :
\(\begin{tikzpicture}[+preamble]\usepackage{forest}[/preamble]\begin{forest}[m 2 ] [ 2] [n 2 ][2][5]]]]]\end{forest}\end{tikzpicture} \)
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Explain why \(11\times13\times15\times17+17 \) is a composite number.
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It has 4 factors so it is a composite number.
Find the LCM and HCF of 120 and 144 by fundamental theorem of arithmetic.
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HCF
LCM
Prove that \(n^2n\) is divisible by \(2 \) for every positive integer \(n \).
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Case i: Let be an even positive integer.
When
In this case , we have
, where
is divisible by
Case ii: Let be an odd positive integer.
When
In this case
, where
is divisible by .
is divisible by for every integer
There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
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Time taken by both to meet again = 2 cm of 18 and 12
LCM
They will meet after 36 minutes
A baker has 444 sweet biscuits and 276 salty biscuits. He wants to stack them in such a way that each stack has the same number and same type of biscuits and they take up the least area of the tray. What is the number of Lbiscuits that can be placed in each stack for this purpose?
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HCF
Total number of biscuits that can be placed in each stack = 12
On a morning walk, three boys step off together and their steps measure 45 cm, 40 cm and 42 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
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LCM
Each person should walk a minimum distance of 2520 cm.
Show that any positive odd integer is of the form \(6q+1\) or \(6q+3\) , where \(q \) is a positive integer.
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For
This will always be an odd integer
For
This will also be an odd integer.
Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is a positive integer.
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For
This will always be an odd integer
For
This will always be an odd integer.
Prove that \(2\sqrt34 \) is an irrational number.
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Let be rational number
…(1)
Let
It has common factor 3 our assumption is wrong. So is an irrational number.
A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sells the oil by filling, the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?
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HCF litres
An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
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HCF
The maximum number of columns
Show that any positive odd integer is of the form 6p + 1, 6p + 3 or 6p + 5, where p is some integer.
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where
For
This will always be an odd integer
For
This will always be an odd integer
For , This will always be an odd integer
Find the LCM and HCF of 15, 18, 45 by the prime factorisation method.
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HCF
LCM
Prove that \(\dfrac{7}{5}\sqrt2 \) is not a rational number.
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Let be a rational number
As is a rational number is also rational which is contradiction because is irrational
So our assumption is wrong is irrational number.