hi, the following is a very difficult HCF problem of arithmetic: PROBLEM: find the greatest number which divides 2300 and 3500 leaving 32 and 56,respectively, as remainders. i tried to find the HCF of these two numbers but it gave 100. which is incorrect.please help me solve this problem. any help would be highly appreciated.
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very difficult HCF problem
- Thread starter friar
- Start date
Hello, friar
Of course it's wrong . . .
You found the greatest number which divides 2300 and 2500 (period!)
. . completely ignoring the input about those remainders.
Denis' explanation is excellent.
Here is my baby-step solution . . .
We are looking for a positive integer \(\displaystyle x\) with two restraints:
1. 2300 equals some integer \(\displaystyle a\) times \(\displaystyle x\) plus the remainder 32.
. . \(\displaystyle 2300 \:=\:ax + 32 \quad\Rightarrow\quad x \:=\:\dfrac{2268}{a} \) .[1]
2. 3500 equals some integer \(\displaystyle b\) times \(\displaystyle x\) plus the remainder 56.
. . \(\displaystyle 3500 \:=\:bx + 56 \quad\Rightarrow\quad x \:=\:\dfrac{3444}{b}\) .[2]
Equate [1] and [2]: .\(\displaystyle \dfrac{2268}{a} \:=\:\dfrac{3444}{b} \quad\Rightarrow\quad 2268b \:=\:3444a\)
. . \(\displaystyle \dfrac{a}{b} \:=\:\dfrac{2268}{3444} \quad\Rightarrow\quad \dfrac{a}{b} \:=\:\dfrac{27}{41} \quad\Rightarrow\quad \begin{Bmatrix}a &=& 27 \\ b &=& 41\end{Bmatrix}\)
Substitute into [1] and [2]:
. . \(\displaystyle x \;=\;\dfrac{2268}{27} \;=\;\dfrac{3444}{41} \;=\;\boxed{84}\)
Find the greatest number which divides 2300 and 3500 leaving 32 and 56, resp., as remainders.
i tried to find the HCF of these two numbers but it gave 100, which is incorrect.
Of course it's wrong . . .
You found the greatest number which divides 2300 and 2500 (period!)
. . completely ignoring the input about those remainders.
Denis' explanation is excellent.
Here is my baby-step solution . . .
We are looking for a positive integer \(\displaystyle x\) with two restraints:
1. 2300 equals some integer \(\displaystyle a\) times \(\displaystyle x\) plus the remainder 32.
. . \(\displaystyle 2300 \:=\:ax + 32 \quad\Rightarrow\quad x \:=\:\dfrac{2268}{a} \) .[1]
2. 3500 equals some integer \(\displaystyle b\) times \(\displaystyle x\) plus the remainder 56.
. . \(\displaystyle 3500 \:=\:bx + 56 \quad\Rightarrow\quad x \:=\:\dfrac{3444}{b}\) .[2]
Equate [1] and [2]: .\(\displaystyle \dfrac{2268}{a} \:=\:\dfrac{3444}{b} \quad\Rightarrow\quad 2268b \:=\:3444a\)
. . \(\displaystyle \dfrac{a}{b} \:=\:\dfrac{2268}{3444} \quad\Rightarrow\quad \dfrac{a}{b} \:=\:\dfrac{27}{41} \quad\Rightarrow\quad \begin{Bmatrix}a &=& 27 \\ b &=& 41\end{Bmatrix}\)
Substitute into [1] and [2]:
. . \(\displaystyle x \;=\;\dfrac{2268}{27} \;=\;\dfrac{3444}{41} \;=\;\boxed{84}\)
very difficult hcf and lcm problems
I should first introduce myself.i am Tanya friend of fria.The following are the exercise questions in a school which I find difficult to solve for children please any body help me solve for the children. Q1) Find the greatest number which divides 2300 and 3500 leaving 32 and 56 respectively, as remainders? Q2) A room is 11m 15 cm long and 8m 91cm broad. The flour of the room is to be paved with square tiles. Find the length of the largest tile that can be used? Q3) Find two numbers between 100 and 150 that have an HCF of 24.? ANSWERS: Q1)84 Q2)81cm Q3) 120 and 144 Q4) find the least number between 200 and 500 which leaves a remainder of 3 in each case when divided by 8, 10, 12 and 30? Q5) find the smallest number which when divided by 12, 15, 18 and 27 leaves as remainder 8,11,14 and 23 respectively? Q6) find the smallest number of 4 digits which when divided by 40, 50 and 60 leaves a remainder of 5 in each case? Q7) Find the greatest number of 5 digits which when divided by 25, 30 and 40 leaves a remainder of 20, 25 and 35 re-spectively? Q8) what is the least number that must be added to 2000 so that the sum is divisible exactly by 10, 12, 16 and 18? Q9) Four bells toll after interval of 8, 9, 12 and 15 minutes, respectively. If they toll together at 3 pm, when will they toll together next? Q10) obtain two numbers such that their HCF is 20 and their LCM is 300? ANSWERS: Q4)243 Q5)536 Q6)1205 Q7)99595 Q8)160 Q9)9 pm Q10) 20 and 300 or 60 and 100.
I should first introduce myself.i am Tanya friend of fria.The following are the exercise questions in a school which I find difficult to solve for children please any body help me solve for the children. Q1) Find the greatest number which divides 2300 and 3500 leaving 32 and 56 respectively, as remainders? Q2) A room is 11m 15 cm long and 8m 91cm broad. The flour of the room is to be paved with square tiles. Find the length of the largest tile that can be used? Q3) Find two numbers between 100 and 150 that have an HCF of 24.? ANSWERS: Q1)84 Q2)81cm Q3) 120 and 144 Q4) find the least number between 200 and 500 which leaves a remainder of 3 in each case when divided by 8, 10, 12 and 30? Q5) find the smallest number which when divided by 12, 15, 18 and 27 leaves as remainder 8,11,14 and 23 respectively? Q6) find the smallest number of 4 digits which when divided by 40, 50 and 60 leaves a remainder of 5 in each case? Q7) Find the greatest number of 5 digits which when divided by 25, 30 and 40 leaves a remainder of 20, 25 and 35 re-spectively? Q8) what is the least number that must be added to 2000 so that the sum is divisible exactly by 10, 12, 16 and 18? Q9) Four bells toll after interval of 8, 9, 12 and 15 minutes, respectively. If they toll together at 3 pm, when will they toll together next? Q10) obtain two numbers such that their HCF is 20 and their LCM is 300? ANSWERS: Q4)243 Q5)536 Q6)1205 Q7)99595 Q8)160 Q9)9 pm Q10) 20 and 300 or 60 and 100.
Sorry
sorry.I should first introduce myself.i am Tanya friend of fria.The following are the exercise questions in a school which I find difficult to solve for children please any body help me solve for the children. Q1) Find the greatest number which divides 2300 and 3500 leaving 32 and 56 respectively, as remainders? Q2) A room is 11m 15 cm long and 8m 91cm broad. The flour of the room is to be paved with square tiles. Find the length of the largest tile that can be used? Q3) Find two numbers between 100 and 150 that have an HCF of 24.? ANSWERS: Q1)84 Q2)81cm Q3) 120 and 144 Q4) find the least number between 200 and 500 which leaves a remainder of 3 in each case when divided by 8, 10, 12 and 30? Q5) find the smallest number which when divided by 12, 15, 18 and 27 leaves as remainder 8,11,14 and 23 respectively? Q6) find the smallest number of 4 digits which when divided by 40, 50 and 60 leaves a remainder of 5 in each case? Q7) Find the greatest number of 5 digits which when divided by 25, 30 and 40 leaves a remainder of 20, 25 and 35 re-spectively? Q8) what is the least number that must be added to 2000 so that the sum is divisible exactly by 10, 12, 16 and 18? Q9) Four bells toll after interval of 8, 9, 12 and 15 minutes, respectively. If they toll together at 3 pm, when will they toll together next? Q10) obtain two numbers such that their HCF is 20 and their LCM is 300? ANSWERS: Q4)243 Q5)536 Q6)1205 Q7)99595 Q8)160 Q9)9 pm Q10) 20 and 300 or 60 and 100.
D
Deleted member 4993
Guest
Friar is an interesting child... G H Hardy might have been a suitable tutor, but, alas, he is dead.
You think Friar is Ramanujam reincarnated??
D
Deleted member 4993
Guest
Friar joined the forum in 12/2011 and posted 4 times in 8 month. So if he wants to be spammer - he is extremely slow one....
May be Robin Hood ousted him from his band of merry men??!!
May be Robin Hood ousted him from his band of merry men??!!
fizza anum
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please explain it
explain itFriar Tuck?
2300 - 32 = 2268
3500 - 56 = 3444
gcd(2268,3444) = 84