Exercise 1.1
Euclid’s division lemma bhut asan hai agr acche se smjh me ajye to. To abb mai explain krna suru krta hu:-
a=bq+r
Dividend(a)=Divisor(b) × Quotient(q) + Remainder(r)
uper image jaise ki dikh raha hai humne pehle :- 1152÷1664 kiya hai jiske baad remainder 512 bcha hai. 512 se hi hum uper wale divisor ko divide krenge aur tb tk krenge jb remainder 0 naa aajaye.jb remainder 0 ata h, tb jo divisor hoga whi hmara hcf rhegaa .hum jitne bar bhi divide krenge utne Baar hum wo ek line likhenge (by using Euclid’s division lemma). to umid hai ki apko smjh me aya hoga.(isme mai ke baat btana bhul gaya😅😅 ki jo hcf ke liye do numbers diye rhte hai usme bde wale ko chote wale se hi divide krte hai.)
Question no. 8 me bola hai ki koi aisa no. jisse agr diye hue dono no. ko divide krte hai to remainder 5 bch rha hai, iska mtlb ki 245 aur 1029 dono me hi remainder 5 bcche ge.kbhi bhi kisi question ko solve krne se pehle usko accha se smjh lete hai taki ye pta chl ske ki question kya kaise krna hai, ek bar agr question acche se smjh me aagya to hum aram se question ko kr skte hai. to abb solution pe ate hai:-
iss question ke hisab se diye gye gye dono no. ko kisi aise no. divide krna hai jisse ki remainder dono ke 5 aye –
to hum sbse pehle dono numbers me 5 minus kr lenge. Ye hum direct minus rough me kr skte hai lekin fair me hume step by step likh ke krna pdega aur exam me bhi taki accche marks mile agr hum kisi question ko krte hai to uske steps pe bhi marks milte hai.
jiss question me remainder diye rhte hai usme aise steps honge
step 1.we have,
step 2. 245 and 1029 leaving remainder 5.
step 3.subtracting 5 in 245 and 1029.(245-5=240 , 1029-5=1024)
step 3.then,
step 4. find the HCF of 240 and 1024.(ye numbers maine minus krne ke baad likhe
hai)
abb iske baad hum normally Euclid’s division lemma ko use krenge.
iss question me pucha hai ki kis no. se hum 1365 ko divide krenge ki 31 quotient ayega(mtlb ki 1365 jitne bar me ktegaa/divide hoga) aur 32 remainder bchega.
to iske liye ekdm normal sa rule hai:-
step 1.We know that
step 2.Dividend=Divisor × Quotient + Remainder
step 3.1365=divisor × 31+32
step 4.divisor × 31=1365-32
step 5.divisor = 1333÷31
step 6.divisor=43
Mai jo bhi explain kr rha hu uske baad ek baar solution dekh lene se aur acche se smjh me ayegaa.😁😁😁
iss question me bataya hai ki jo 408 aur 1032 ka HCF hai wo 1032m – 408×5 ke equal hai mtlb ki:-
HCF( 408,1032)=1032m-408 × 5
to iske hisab se normally hum pehle ( 408,1032) kaa HCF nikal lenge pehle to solution apko copy me dikh rha hoga usi trike se krna hai bs normal saa euclid’s division lemma fir se lga dena hai bs uske baad, jo bhi HCF ayega usko hum ” HCF( 408,1032) ” iski jgh rkhenge aur normally jaise hum dushre questions me “x” ki value find krte the waise hi hum isme m ki value find krenge, to fir age bdhte hai:-
HCF( 408,1032)=1032m-408 × 5
24= 1032m-408 × 5
24=1032m-2040
1032m-2040=24 (humne dono numbers ka side change kr diya hai taki solution krne
me koi problem naa ho)
1032m=24+2040
1032m=2064
m=2064÷1032
m=2
To dikh rha hoga ye bilkul normal value nikalne jaisa hi hai to agr koi bhi problem ho to comment kre.aur sare questions solve krne se pehel euclid’s lemma division acche se smjh le pdh le taki aur asani ho.😁😁😁✌
Iss question me jade kuch hard nhi hai bs isme alg ye hai ki isme do remainder de diya hai, isse confuse hone ki jrurt nhi hai kyuki question age hum dhyan se pdhe to usme last me likha hoga respectively(क्रमश:) iska mtlb hua ki bari bari se, isse hume ye pta chlta hai ki jo do remainder aur do number diye hue hai unko hume ek ek krke minus(-) krna hai jaise isme pehla number 2053 diya hai aur dushra number 967 hai. Remainder me pehla number 5 aur dushra number 7 hai. To jaise respectively likha hua hai question me last me uske hosab se hum pehle 2053-5 krenge aur dushra 967-7 krenge aur uske baad jo minus krne ke baad jo number ayega uska hum HCF nikal lenge Euclid’s division lemma se jo ki apko suruat me hi bataya gaya hai, aur iske baad jo iss explaition se pehle question no. 16 ka solution hai agr hum ussme dekehe to aur acche se smjh aajaye gaa.😁😁
iss question me pucha hai ki agr d 56 aur 72 kaa hcf hai to fir x aur y kaa value find kro?
iss trike ke question me mai khud fsa tha lekin kai bar solve krne aur help lene ke baad mujhe bhi smjh agaya hai. Isme bhi hume value hi find krna hai lekin jaisa ki question me dikh rha hoga isme do value find krna hai x & y to isme pehle hum hur question ki trh HCF nikal lenge uske baad fir uske age ka mai suru krta hu. Jisa ki app dekh rhe honge question me “d=56x+72y” diya hua hai aur apko question pdhne se pta chl gaya hai ki d jo hai wo 56 aur 72 ke hcf ke equal hai to hum kh skte hai ki “HCF(56,72)= d=56x+72y”.
isme hum jb HCF nikalenge to usme equation bhi bna lenge jisse ki age asani ho:-
By using Euclid’s divisions lemma:-
72=56×1+16 -②
Again,By using Euclid’s divisions lemma:-
56=16×3+8 -①
Again,By using Euclid’s divisions lemma:-
16=8×2+0
HCF(56,72)=8
d=8
d=56x+72y
8=56x+72y (abb jaisa ki dikh raha hai ki yaha do no. hai jinke value find krna hai to hume koi trika nikalna hoga jisse ki hum isko solve kr paye, to hum dekhenge ki 8 hai uski koi aur value askta hai kya to hum ek bar fir uperse apne solution me dekhenge ki khi mil jaye uska value aur usme in dono me so koi number bhi hona chahiye taki solution asan ho jaye)
8=56-3×16 (from equation ①, to apko lg rha hoga ki ye no. to equation first me hai nhi, agr dhyan se dekhe to humne bs number ka jgh exchange kr diya gaya hai app equation ① me dekh skte hai )
8=56-3×(72-56) [from equation ②, to isme jaisa ki dikh raha hai humne 16 ki jgh inn numbers ko rkh liya hai to agr hum equation ② ko dekhe to apko ye condition bhi bilkul uper wale ki trh hi lgega ]
8=56-72×3+56×3 (isme apko saf dikh rha hai ki bracket ke bahar tha -3 to hum use bracket ke ander wale se multiply kr denge aur sign kaa hum acche se dhyan rkhenge nhi to ans glt bhi asakta hai)
8=56×(+4)+72×(-3) [isme humne 56 ko 4 se isiliye multiply krte hue dikhaya hai kyuki uper wale equation me dikh rha tha ki 3 ki humne into kiya tha 56 ko aur suruat me bhi 56 tha to do jgh rkhne se accha hai ki hum use ek hi jgh rkhe aur humne -3 ko bracket me rkh kr 72 se into kr diya hai]
d=56x+72y (d=8,x=+4,y=-3. abb thora confusion hoga ki kaise aya hai x aur y ka value to dhyan se uper wale equation me dekhe ki uper wale equation aur jo abhi hmne equation likha hai wo milta julta nhi hai, isiliye humne -3 ko bracket me likhe the hum log)
iske age kaise apko copy me represent krna hai ye chiz apko jis page me solution hai usme dikh jayegaa.
isme abhi kuch questions hai jo mujhe bhi nhi arhe hai😅😅
to wo mai baad me smjha dunga uske liye app bde wale copy me just 4 pages chor denge okk.
EXERCISE 1.2
Class 9 Notes 