Explain why 13233343563715 is a composite number?

1

(a)

The given number ends in 5. Hence it is a multiple of 5. Therefore it is a composite number

What is the HCF of the smallest composite number and the smallest prime number?

1

(a)

The smallest prime number is 2 and the smallest composite number is 2²
Hence, required HCF (2²,2) = 2

Calculate the HCF of 3³ x 5 and 3² x 5².

1

(a)

HCF of 3³ x 5 and 3² x 5²
= 3² x 5
= 9 x 5
= 45

What is the condition for the decimal expansion of a rational number to terminate? Explain with the help of an example.

1

(a)

The decimal expansion of a rational number terminates, if the denominator of rational no p/q, when p and q are co-primes and q can be expressed as 2^m5ⁿ  where In and n are non-negative integers.
e.g. \(\frac { 3 }{ 10 } =\frac { 3 }{ 2^{ 1 }\times 5^{ 1 } } =0.3\)

Calculate \(\frac { 3 }{ 8 } \) in the decimal form.

1

(a)

\(\frac { 3 }{ 8 } =\frac { 3 }{ { 2 }^{ 3 } } \)
    \(=\frac { { 3\times 5 }^{ 3 } }{ { 2 }^{ 3 }\times { 5 }^{ 3 } } \)
    \(=\frac { 375 }{ 10^{ 3 } } =\frac { 375 }{ 1000 } \)
   \(=0.375\)

Write the HCF and LCM of the smallest odd composite number and the smallest odd prime number. If an odd number p divides q², then will it divide q³ also? Explain.

2

(a)

Find the LCM of x and y, if xy = 180 and HCF of (x, y) = 5

2

(a)

Find the least number that is divisible by all the numbers from 1 to 5 (both inclusive).

2

(a)

Find the missing numbers a, b, c and d in the given factor tree

2

(a)

Explain whether \(3\times 12\times 101+4\)  is a prime number or a composite number

2

(a)

Show that any positive even integer can be written in the from 6q, 6q + 2 or 6q + 4, where q is an integer

2

(a)

Show that 7ⁿ cannot end with the digit zero, for any natural number II.

2

(a)

Find the HCF of 960 and 432.

3

(a)

Explain, why (3 x 5 x 7) + 7 is a composite number?

3

(a)

If HCF of two numbers is 2 and their product is 120, find their LCM.

3

(a)

Find the greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5, respectively.

3

(a)

Use Euclid division lemma to show that the square of any positive integer cannot be of the form 5m + 2 or 5m + 3 for some integer m.

3

(a)

1. The product of two consecutive positive integers is divisible by 2. Is this statement true or false? Give reason.

   3

   (a)

2. Show that the square of an odd positive integer is of the form 8m + 1, where m is some whole number.

   3

   (a)

1. If two positive integers p and q can be expressed as p = ab² and q = a³b; where a,b being prime numbers, find the LCM (p,q)

   3

   (a)

2. The HCF of two numbers is 113 and their LCM is 56952. If one number is 904, find the other number.

   3

   (a)

1. Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1, for some integer m.

   5

   (a)

2. Explain, why (7 x 11 x 13) + 13 and (7 x 6 x 5 x 4 x 3 x 2 x 1) + 5 are composite numbers?

   5

   (a)

1. If the HCF of 657 and 963 is expressible in the form of 657x + 963 (-15), find x.

   5

   (a)

2. The traffic lights at three different roads crossings change after 48, 72 and 108 s, respectively. If they change simultaneously at 7 am, then at what time will they change simultaneously again?

   5

   (a)

1. Write the missing numbers is the following factorisation.
   (i)       (ii) 

   10

   (a)

2. For any positive integer n, prove that n³ - n is divisible by 6.

   10

   (a)