Two positive integers p and q can be expressed as = ab2 and q =a2b, a and b are prime numbers. what is the LCM Of p and q ?
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Find the LCM and HCF of the following pairs of integer and verify that LCM×HCF=Product of the two numbers.
Solution : i) 26 and 91 26=2×13×1(expressing as product of it’s prime factors) 91=7×13×1(expressing as product of it’s prime factors) So, LCM(26,91)=2×7×13×1=182 HCF(26,91)=13×1=13 Verification: LCM×HCF=13×182=2366 Product of 26 and 91 =2366 Therefore,LCM×HCF=Product of the two numbers . i) 510 and...
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Find the LCM and HCF of the following integers by applying the prime factorization method. i) 12,15 and 21 ii) 17,23 and 29 iii) 8,9 and 25
Solution : i) 12,15 and 21 12=2×2×3 15=5×3 21=7×3 From the above ,HCF(12,15,21)=3and LCM(12,15,21)=420 ii)17,23,and 29 17=17×1 23=23×1 29=29×1 From the above ,HCF(17,23,29)=1and LCM(17,23,29)=11339 iii)8,9 and 25 8=2×2×2 9=3×3 25=5×5 From the above ,HCF(8,9,25)=1and LCM(8,9,25)=1800
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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
Correct answer is option (C) 0 ≤ r < b
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If two positive integers x and y are expressible in terms of primes as x = p^2q^3 and y = p^3q, what can you say about their LCM and HCF.
Solution: x = p2q3 and y = p3q HCF(x, y) = p2 x q LCM(x, y) = p3 x q3 LCM(x, y) = (p x q2) x HCF(x, y) => LCM is a multiple of HCF.
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If two positive integers a and b are written as a = x^3y^2 and b = xy^3; x, y are prime numbers, then HCF (a, b) is
Correct answer is (B) xy2
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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 p/q ?
Solution: 327.7081 = 3277081/10000 = 3277081 / 24 x 54 = p/q Here, q is of the form 2m × 5n, where m and n are natural numbers. The prime factors of p and q will...
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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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Define HOE of two positive integers and find the HCF of the following pairs of numbers: (i) 32 and 54 (ii) 18 and 24 (iii) 70 and 30 (iv) 56 and 88
By applying Euclid’s division lemma (i) 54 = 32 × 1 + 22 Since remainder ≠ 0, apply division lemma on division of 32 and remainder 22. 32 = 22 × 1 +...
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If a is an odd number, b is not divisible by 3 and LCM of a and b is P, what is the LCM of 3a and 2b?
The correct answer is 6p
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The average of 5 positive integers is 436. The average of the first two numbers is 344 and the average
(d) Third number = (436 × 5) – (344 × 2 + 554 × 2) = 2180 – 1796 = 384
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