Euclid’s division lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy
(A) 1 < r < b (B) 0 < r ≤ b
(C) 0 ≤ r < b (D) 0 < r < b
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Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.
Solution: Let the two consecutive odd positive integers be x and x + 2. x < 10, x + 2 < 10, x + (x + 2) > 11 x + 2 <...
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Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and (i) deg p(x) = deg q(x) (ii) deg q(x) = deg r(x)
Solution: According to the division algorithm, if p(x) and g(x) are two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) ×...
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Use Euclid’s division algorithm to find HCF of 441, 567, 693.
Solution: By Euclid’s division algorithm, 693 = 567 x 1 + 126 567 = 126 x 4 + 63 126 = 63 x 2 + 0 So, HCF(441, 63) = 63 So, HCF (693, 567) =...
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Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.
Solution: Since, 1, 2 and 3 are the remainders of 1251, 9377 and 15628 respectively. So, 1251 – 1 = 1250 is exactly divisible by the required number, 9377 – 2 = 9375...
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Prove that one of any three consecutive positive integers must be divisible by 3.
Solution: Let the three consecutive positive integers be n, n + 1 and n + 2, where n is any integer. By Euclid’s division lemma, we have a = bq + r; 0 ≤...
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For any positive integer a and 3, there exist unique integers q and r such that a = 3q + r, where r must satisfy
Correct answer is (A) 0 ≤ r < 3
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If a and b are two odd positive integers such that a > b, then prove that one of the two numbers (a+b)/2 and (a-b)/2 is odd and the other is even.
Let, a = 2p+1, p∈N and b = 2q+1 , q∈N\(\cup\){0} \(\because\) a > b ⇒ 2p+1 > 2q+1 ⇒ p > q \(\frac{a+b}{2}\) = \(\frac{2p+1+2q+1}{2}\) = p+q+1 \(\frac{a-b}{2}\) = \(\frac{2p+1-(2q+1)}{2}\) = p-q Case-I: \(\frac{a+b}{2}\) is odd which implies p+q+1 is odd \(\Rightarrow\) p+q is even (\(\because\) odd - 1 = even) \(\Rightarrow\) p+q-2q...
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Using Euclid’s division algorithm, check whether 231, 396 are co-prime or not. Justify your answer.
396 = 1 × 231 + 165 231 = 1 × 165 + 66 165 = 2 × 66 + 33 66 = 2 × 33 + 0 ∴ The HCF of 231, 396...
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There are 15 dozen candles in a box. If there are 39 such boxes. How many candles are there in all the boxes together?
(a) Total number of candles = 15 × 12 × 39 = 7020
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There are 12 dozen mangoes in a box. If there are 43 such boxes, how many mangoes are there in all the boxes together?
(c) Number of mangoes = 12 dozens = 12 × 12 = 144 Number of mangoes in 43 boxes = 43 × 144 = 6192
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