Explain why 13233343563715 is a composite number?

a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then calculate the least prime factor of (a + b).

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

Find the smallest positive rational number by which 1/7 should be multiplied so that its decimal expansion terminates after 2 places of decimal.

The decimal representation of  \(\\ \frac { 6 }{ 1250 } \)  will terminate 1250 after how many places of decimal?

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 \(\frac{p}{q}\)? Give reason.

Given that HCF (306, 1,314) = 18. Find LCM (306, 1,314).

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

An army contingent of 104 members is to march behind an army band of 96 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?

By using Euclid's division algorithm, find the largest number which when divides 969 and 2059, gives the remainders 9 and 11, respectively.

The numbers 525 and 3000 are both divisible by 3, 5, 15, 25 and 75. What is the HCF of 525 and 3000? Justify your answer.

Find the HCF and LCM of 510 and 92 and verify that HCF x LCM = Product of two given numbers

Prove that \(\sqrt { 2 } \)   is an irrational .

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

In Euclid's division lemma a = bq + r, where \(0 \leq r<b\) . What is a?

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?

Prove that, if a, b, c and d are positive rationals such that \(a+\sqrt { b } =c+\sqrt { d } \), then either a = c and b = d or b and d are squares of rationals.

Can the number 6ⁿ, n being number, end with the digit 5 ? Give reasons.