HCF Calculation for Powers of 2
The problem asks us to find the Highest Common Factor (HCF) of two numbers expressed in the form \(2^n - 1\). Specifically, we need the HCF of \(2^{36}-1\) and \(2^{45}-1\). Notice that both numbers are generated using the same base, 2, but with different exponents (36 and 45).
Explaining the HCF Property for Powers
There's a useful mathematical property for finding the HCF of numbers of the form \(a^n - 1\) and \(a^m - 1\). The property states:
HCF(\(a^n - 1\), \(a^m - 1\)) = \(a^{\text{HCF}(n, m)} - 1\)
This means we can find the HCF of the original numbers by first finding the HCF of their exponents and then plugging that result back into the same form (\(a^{\text{HCF}}-1\)).
HCF Step-by-Step Calculation
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Identify Components:
In our problem, the base is \(a=2\). The exponents are \(n=36\) and \(m=45\).
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Calculate HCF of Exponents:
First, let's find the HCF of the exponents, 36 and 45.
We can list the factors of each number:
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Factors of 45: 1, 3, 5, 9, 15, 45
The common factors are 1, 3, and 9. The highest common factor is 9.
So, HCF(36, 45) = 9.
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Apply the Property:
Now we use the property HCF(\(a^n - 1\), \(a^m - 1\)) = \(a^{\text{HCF}(n, m)} - 1\).
Substitute \(a=2\) and HCF(36, 45) = 9:
HCF(\(2^{36}-1\), \(2^{45}-1\)) = \(2^{\text{HCF}(36, 45)} - 1\)
= \(2^9 - 1\)
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Compute the Result:
Finally, calculate the value of \(2^9 - 1\).
We know that \(2^9 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 512\).
Therefore, \(2^9 - 1 = 512 - 1 = 511\).
Final HCF Result
The HCF of \(2^{36}-1\) and \(2^{45}-1\) is 511.
Similar Questions
- If HCF of 768 and \(x^6y^2\) is \(32xy\) for natural numbers \(x \ge 2, y \ge 2\), then what is the value of \((x + y)\)?
The HCF of \(x\) and \(y\) is \(H\). Consider the following statements in respect of the HCF of \(p=\frac{x^3 + y^3}{x^2-xy+y^2}\) and \(q=\frac{x^3-y^3}{x^2+xy+y^2}\) :
I. The HCF of \(p\) and \(q\) can be \(H\).
II. The HCF of \(p\) and \(q\) can be \(2H\).
Which of the statements given above is/are correct?
- What is the HCF of \(x^3 + y^3 + 3xy-1\) and \((x + y)^2 - 1\)?
- A number N is such that when divided by 4, 6, 7 or 9, it leaves 3 as remainder. What is the smallest 4-digit number that satisfies this property?
- A plank of wood 4·25 m long and 3·4 m wide is to be cut into square pieces of equal size. How many square pieces of largest size can be cut from the plank, if no wastage is allowed ?
- What is the HCF of \(x^4 - 13x^2y^2 - 300y^4\), \(x^3 - 4x^2y - 4xy^2 - 5y^3\) and \(x^3 - 125y^3\)?
- Let \(N\) be a 5-digit number. When \(N\) is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of \(N\)?
Let \(N\) be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of he following is correct?
Let \(n\) be a natural number. The HCF of \(n, n + 10\) is 10. If the LCM is \(x\) (a 2-digit number), then how many values of \(x\) are possible?
Consider the following statements in respect of prime numbers p and q:
- I. Their LCM is always an odd number.
- II. Sum of their LCM and HCF is always an even number.
Which of the statements given above is/are correct?
Important Questions from LCM and HCF
The HCF and LCM of two numbers are 12 and 72, respectively. If the ratio of the two numbers is 2 ∶ 3, then the larger of the two numbers is:
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
Joseph visits the club on every 5 th day, Harsh visits on every 24 th day, while Sumit visits on every 9 th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
What is the least number which when divided by 12,20 and 24 leaves in each case a remainder of 8?
Which of the following is a pair of co-primes?
