Real Numbers Questions

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 »

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 »

Question : Express each number as a product of its prime factors : (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429 Solution : (i)  We use the division method as shown below : \(\therefore\)   140 = \(2 \times 2 \times 5 \times 7\) = \(2^2 \times 5 \times 7\) (ii)  We use the division …

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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 »

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 »

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.

Solution : By Euclid’s division algorithm, we have a = bq + r                   ……….(i) On putting, b = 6 in (1), we get a = 6q + r             [0 \(\le\) r < 6] If r = 0, a = 6q, 6q is …

Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer. Read More »

Question : Use Euclid’s division algorithm to find the H.C.F of : (i) 135 and 225 (ii) 196 and 38220 (iii) 865 and 225 Solution : (i) We start with the larger number 225. By Euclid’s Division Algorithm, we have 225 = 135 \(\times\) 1 + 90 We apply Euclid’s Division Algorithm on Division 135 …

Use Euclid’s division algorithm to find the H.C.F of : Read More »