Math-_Class_10th_-_Number_System_with_Solutions

Apr 30th, 2015
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Numbers are intellectual witnesses that belong only to mankind. 1. If the H C F of 657 and 963 is expressible in the form of 657x + 963x - 15 find x. (Ans:x=22)

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Math- Class 10th - Number System with SolutionsNUMBER SYSTEMSNumbers are intellectual witnesses that belong only to mankind. 1. If the H C F of 657 and 963 is expressible in the form of 657x + 963x - 15 find x. (Ans:x=22) Ans: Using Euclids Division Lemma a= bq+r , o r < b 963=6571+306 657=3062+45 306=456+36 45=361+9 36=94+0 HCF (657, 963) = 9 now 9 = 657x + 963 (-15) 657x=9+96315 =9+14445 657x=14454 x=14454/657 2. Express the GCD of 48 and 18 as a linear combination. (Ans: Not unique) A=bq+r, where o r < b 48=18x2+12 18=12x1+6 12=6x2+0 HCF (18,48) = 6 now 6= 18-12x1 6= 18-(48-18x2) 6= 18-48x1+18x2 6= 18x3-48x1 6= 18x3+48x(-1) i.e. 6= 18x +48y x =22 x=3 , y=-1 76= 183 +48(-1) =183 +48(-1) + 1848-1848 =18(3+48)+48(-1-18) =1851+48(-19) 6=18x+48y Hence, x and y are not unique. 3. Prove that one of every three consecutive integers is divisible by 3. Ans: n,n+1,n+2 be three consecutive positive integers We know that n is of the form 3q, 3q +1, 3q + 2 So we have the following cases Case I when n = 3q In the this case, n is divisible by 3 but n + 1 and n + 2 are not divisible by 3 Case - II When n = 3q + 1 Sub n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not divisible by 3 Case III When n = 3q +2 Sub n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not divisible by 3 Hence one of n, n + 1 and n + 2 is divisible by 3 4. Find the lar

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