For some integer m, every even integer is of the form
\(m\)
\(m + 1\)
\(2m\)
\(2m + 1\)
For some integer \(q\), every odd integer is of the form
\(q\)
\(q + 1\)
\(2q\)
\(2q + 1\)
\(n^2 – 1\) is divisible by 8, if \(n\) is
an integer
a natural number
an odd integer
an even integer
If the HCF of 65 and 117 is expressible in the form \(65m – 117\), then the value of \(m\) is
4
2
1
3
The largest number which divides 70 and 125, leaving remainders 5 and 8, respectively, is
13
65
875
1750
If two positive integers a and b are written as \(a = x^3y^2\) and \(b = xy^3\); \(x, y\) are prime numbers, then \(HCF(a, b)\) is
\(xy\)
\(xy^2\)
\(x^3y^3\)
\(x^2y^2\)
If two positive integers p and q can be expressed as \(p = ab^2\) and \(q = a^3b\); \(a, b\) being prime numbers, then \(LCM(p, q)\) is
\(ab\)
\(a^2b^2\)
\(a^3b^2\)
\(a^3b^3\)
The product of a non-zero rational and an irrational number is
always irrational
always rational
rational or irrational
one
The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is
10
100
504
2520
The decimal expansion of the rational number \(\dfrac{14587}{1250}\) will terminate after:
one decimal place
two decimal places
three decimal places
four decimal places
Write whether every positive integer can be of the form \(4q+2\), where \(q\) is an integer. Justify your answer.
No, not every positive integer can be of the form \(4q+2\).
“The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.
True. The product of two consecutive positive integers is always divisible by 2.
“The product of three consecutive positive integers is divisible by 6”. Is this statement true or false? Justify your answer.
The statement is True. The product of three consecutive positive integers is always divisible by 6.
Write whether the square of any positive integer can be of the form \(3m+2\), where \(m\) is a natural number. Justify your answer.
No.
A positive integer is of the form \(3q+1\), q being a natural number. Can you write its square in any form other than \(3m+1\)? Justify your answer.
No. The square of such a number is always of the form \(3m+1\).
The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer.
HCF = 75.
Explain why \(3 \times 5 \times 7 + 7\) is a composite number.
\(3 \times 5 \times 7 + 7 = 112\), and since \(112 = 7 \times 16\), it has more than two factors. Hence it is a composite number.
Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.
No, such numbers cannot exist because \(18\) does not divide \(380\).
Without actually performing the long division, find if \(\dfrac{987}{10500}\) will have terminating or non-terminating decimal expansion. Give reasons.
Terminating decimal expansion, because denominator reduces to \(2^2 \times 5^3\).
A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of \(q\), when this number is expressed in the form \(\dfrac{p}{q}\)? Give reasons.
Since 327.7081 is a terminating decimal number, \(q\) must be of the form \(2^m 5^n\), where \(m,n\) are natural numbers (non-negative integers).
Show that the square of any positive integer is either of the form \(4q\) or \(4q + 1\) for some integer \(q\).
Squares are of the form \(4q\) or \(4q+1\).
Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
Cubes are of the form \(4m\), \(4m+1\) or \(4m+3\).
Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.
Impossible to be \(5q+2\) or \(5q+3\).
Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.
Impossible to be \(6m + 2\) or \(6m + 5\).
Show that the square of any odd integer is of the form 4q + 1, for some integer q.
\(4q + 1\)
If n is an odd integer, then show that \(n^2 - 1\) is divisible by 8.
Divisible by 8.
Prove that if \(x\) and \(y\) are both odd positive integers, then \(x^2 + y^2\) is even but not divisible by \(4\).
Even, not divisible by 4.
Use Euclid’s division algorithm to find the HCF of 441, 567, 693.
63
Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.
625
Prove that \(\sqrt{3}+\sqrt{5}\) is irrational.
Irrational.
Show that 12n cannot end with the digit 0 or 5 for any natural number n.
Cannot end with 0 or 5.
On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
2520 cm (i.e., 25.2 m)
Write the denominator of the rational number \(\dfrac{257}{5000}\) in the form 2m × 5n, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.
Denominator: \(2^3 \cdot 5^4\). Decimal: 0.0514
Prove that \(\sqrt{p}+\sqrt{q}\) is irrational, where \(p, q\) are primes.
Irrational.
Show that the cube of a positive integer of the form \(6q + r\), where \(q\) is an integer and \(r = 0, 1, 2, 3, 4, 5\), is also of the form \(6m + r\).
Yes. The cube of such an integer is again of the form \(6m + r\).
Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
Exactly one out of n, n + 2 and n + 4 is divisible by 3.
Prove that one of any three consecutive positive integers must be divisible by 3.
Yes, in every set of three consecutive integers, one of them is divisible by 3.
For any positive integer \(n\), prove that \(n^3 - n\) is divisible by 6.
Yes. For every positive integer \(n\), the number \(n^3 - n\) is divisible by \(6\).
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.
Exactly one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5.