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).