Answer
Verified
378.9k+ views
Hint: H.C.F can be defined as the highest number which divides the given numbers and gives remainder zero. Apply this rule to get an answer.
Let us suppose that number is ‘N’ which divides 2053 and 967 and leaves a remainder 5 and 7 respectively.
Now, the first number is given as 2053 which is divided by N and given remainder 5. Now, if we subtract 5 from 2053, we get 2053 – 5 = 2048 which will give remainder zero if we divide it by N now.
For second number 967, if we divide it by N, we get remainder 7 which means if we divide 967 – 7 =960 by N, we will get remainder zero.
Hence, now one can rewrite the question as evaluating the number which divides 2048 and 960 and gives remainder 0.
Now, we can relate that number N should be the highest common factor of 2048 and 960 which will divide the given numbers and the remainder will be zero.
So, let us find H.C.F by prime factorization method.
Prime factorization of 960 can be given as;
Hence,
$960=2\times 2\times 2\times 2\times 2\times 2\times 3\times 5$ …………………..(1)
Now, factoring 2048 as
Hence, 2048can be expressed as;
$2048=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$ ………………..(2)
Now, as we know the prime factorization method that the terms common in the factorization of numbers will give the HCF.
Hence, common factors in 960 and 2048;
\[H.C.F=2\times 2\times 2\times 2\times 2\times 2={{2}^{6}}\]
H.C.F = 64
Hence, number N is 64 which will give remainders 5 and 7 on dividing 2053 and 967 respectively.
Note: One can go wrong after subtraction of remainders from the given numbers. He/she may go for calculation of L.C.M (Least common multiple) in place of H.C.F which will give the wrong solution.
Hence, terms H.C.F and L.C.M should be very clear for this question.
One can miss any prime factor from the expression of prime factorization of 960 and 2048.
Let us suppose that number is ‘N’ which divides 2053 and 967 and leaves a remainder 5 and 7 respectively.
Now, the first number is given as 2053 which is divided by N and given remainder 5. Now, if we subtract 5 from 2053, we get 2053 – 5 = 2048 which will give remainder zero if we divide it by N now.
For second number 967, if we divide it by N, we get remainder 7 which means if we divide 967 – 7 =960 by N, we will get remainder zero.
Hence, now one can rewrite the question as evaluating the number which divides 2048 and 960 and gives remainder 0.
Now, we can relate that number N should be the highest common factor of 2048 and 960 which will divide the given numbers and the remainder will be zero.
So, let us find H.C.F by prime factorization method.
Prime factorization of 960 can be given as;
Hence,
$960=2\times 2\times 2\times 2\times 2\times 2\times 3\times 5$ …………………..(1)
Now, factoring 2048 as
Hence, 2048can be expressed as;
$2048=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$ ………………..(2)
Now, as we know the prime factorization method that the terms common in the factorization of numbers will give the HCF.
Hence, common factors in 960 and 2048;
\[H.C.F=2\times 2\times 2\times 2\times 2\times 2={{2}^{6}}\]
H.C.F = 64
Hence, number N is 64 which will give remainders 5 and 7 on dividing 2053 and 967 respectively.
Note: One can go wrong after subtraction of remainders from the given numbers. He/she may go for calculation of L.C.M (Least common multiple) in place of H.C.F which will give the wrong solution.
Hence, terms H.C.F and L.C.M should be very clear for this question.
One can miss any prime factor from the expression of prime factorization of 960 and 2048.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE