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 ?

Solution : They will be again at the starting point at least common multiples of 18 and 12 minutes. To find the L.C.M of 18 and 12, we have : 18 = \(2 \times 3\times 3\) and  12 = \(2 \times 2 \times 3\) L.C.M of 18 and 12 = \(2 \times 2 \times 3 …

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 ? Read More »

Explain why \(7 \times 11 \times 13\) + 13 and \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) + 5 are composite numbers.

Solution : We have,  \(7 \times 11 \times 13\) + 13 = 1001 + 13 =1014 1014 = \(2 \times 3 \times 13 \times 13\) So, it is the product of more than two prime numbers. 2, 3 and 13. Hence, it is a composite number. \(7 \times 6 \times 5 \times 4 \times 3 …

Explain why \(7 \times 11 \times 13\) + 13 and \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) + 5 are composite numbers. Read More »

Check whether \(6^n\) can end with the digit 0 or any n \(\in\) N.

Solution : If the number \(6^n\) ends with the digit zero. Then it is divisible by 5. Therefore the prime factorisation of \(6^n\) contains the prime 5. This is not possible because the only primes in the factorisation of \(6^n\) are 2 and 3 and the uniqueness of the fundamental theorem of arithmetic guarantees that …

Check whether \(6^n\) can end with the digit 0 or any n \(\in\) N. Read More »

Find the H.C.F and L.C.M of the following integers by applying prime factorisation method.

Question : Find the H.C.F and L.C.M of the following integers by applying prime factorisation method. (i)  12, 15, 21 (ii)  17, 23, 29 (iii)  8, 9 and 25 Solution : (i) 12, 15, 21 12 = \(2 \times 2 \times 3\) 15 = \(3 \times 5\) 21 = \(3 \times 7\) Here 3 is a …

Find the H.C.F and L.C.M of the following integers by applying prime factorisation method. Read More »

Find the L.C.M and H.C.F of the following pairs of integers and verify :

Question : Find the L.C.M and H.C.F of the following pairs of integers and verify : L.C.M \(\times\) H.C.F = Product of the two numbers (i)  26 and 91 (ii)  510 and 92 (iii)  336 and 54 Solution :  (i)  26 and 91 26 = 2 \(\times\) 13        and       91 …

Find the L.C.M and H.C.F of the following pairs of integers and verify : Read More »

Use Euclid’s Division Lemma to show that the cube of any positive integer is either of the form 9m or 9m + 1 or 9m + 8.

Solution : Let m be any positive integer. Then it is of the form 3m, 3m + 1 or 3m + 2. Now, we have to prove that the cube of these can be rewritten in the form 9q, 9q + 1 or 9q + 8. Now, \((3m)^3\) = \(27m^3\) = \(9(m^3)\) = 9q, where …

Use Euclid’s Division Lemma to show that the cube of any positive integer is either of the form 9m or 9m + 1 or 9m + 8. Read More »

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.

Solution : By Euclid’s Division Algorithm, we have a = bq + r       …………..(i) On putting b = 3 in (1), we get a = 3q + r,      [0 \(\le\) r < 3] If r = 0   a = 3q  \(\implies\)  \(a^2\) = \(9q^2\)                …

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. Read More »

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 ?

Solution : To find the maximum number of columns, we have to find the H.C.F. of 616 and 32 i.e. 616 = 32 \(\times\) 19 + 8 and 32 = 8 \(\times\) 4 + 0 \(\therefore\) H.C.F of 616 and 32 is 8. Hence, maximum number of columns is 8.