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?
Finding the Least Positive Multiple N with Specific Conditions
The problem asks us to find the least positive integer, let's call it N, that satisfies two conditions:
- It is a multiple of 11.
- It leaves a remainder of 5 when divided by 6, 12, 15, and 18.
We also need to determine which range the value of N falls into.
Step 1: Analyze the Remainder Condition
The condition that N leaves a remainder of 5 when divided by 6, 12, 15, and 18 can be expressed using modular arithmetic:
- \(N \equiv 5 \pmod{6}\)
- \(N \equiv 5 \pmod{12}\)
- \(N \equiv 5 \pmod{15}\)
- \(N \equiv 5 \pmod{18}\)
If we subtract 5 from N, the result (\(N-5\)) must be perfectly divisible by 6, 12, 15, and 18. This means (\(N-5\)) is a common multiple of these numbers.
To find the *least* such positive number N, we first need the Least Common Multiple (LCM) of 6, 12, 15, and 18.
Step 2: Calculate the Least Common Multiple (LCM)
Let's find the prime factorization of each number:
- \(6 = 2 \times 3\)
- \(12 = 2^2 \times 3\)
- \(15 = 3 \times 5\)
- \(18 = 2 \times 3^2\)
The LCM is found by taking the highest power of each prime factor present:
LCM\((6, 12, 15, 18) = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180\).
Step 3: Express N based on the LCM
Since (\(N-5\)) must be a multiple of the LCM (180), we can write:
\(N - 5 = 180k\), where \(k\) is a positive integer.
Rearranging this, we get the general form of N:
\(N = 180k + 5\).
Step 4: Apply the Multiple of 11 Condition
We are looking for the *least positive multiple of 11*. So, N must satisfy:
\(N \equiv 0 \pmod{11}\).
Substitute the expression for N from Step 3:
\(180k + 5 \equiv 0 \pmod{11}\).
Now, let's simplify the coefficient 180 modulo 11:
\(180 \div 11 = 16\) with a remainder of \(4\). So, \(180 \equiv 4 \pmod{11}\).
The congruence becomes:
\(4k + 5 \equiv 0 \pmod{11}\).
Subtract 5 from both sides:
\(4k \equiv -5 \pmod{11}\).
Since \(-5 \equiv 6 \pmod{11}\), we have:
\(4k \equiv 6 \pmod{11}\).
To solve for \(k\), we need to find the multiplicative inverse of 4 modulo 11. We can test values: \(4 \times 1 = 4\), \(4 \times 2 = 8\), \(4 \times 3 = 12 \equiv 1 \pmod{11}\). The inverse is 3.
Multiply both sides of the congruence by 3:
\(3 \times (4k) \equiv 3 \times 6 \pmod{11}\)
\(12k \equiv 18 \pmod{11}\)
Simplify using modulo 11:
\(1k \equiv 7 \pmod{11}\)
\(k \equiv 7 \pmod{11}\).
The smallest positive integer value for \(k\) is 7.
Step 5: Calculate the Value of N
Substitute \(k=7\) back into the equation for N:
\(N = 180k + 5 = 180(7) + 5 = 1260 + 5 = 1265\).
Let's verify:
- Is 1265 a multiple of 11? \(1265 \div 11 = 115\). Yes.
- Does 1265 leave a remainder of 5 when divided by 6, 12, 15, 18? Yes, as it's of the form \(180k + 5\).
- Is it the *least* positive multiple? Yes, because we used the smallest positive integer \(k=7\).
Step 6: Determine the Correct Range
We found \(N = 1265\). Now we check the given options:
- \(900 < N < 1000\): \(900 < 1265 < 1000\) is false.
- \(1000 < N < 1100\): \(1000 < 1265 < 1100\) is false.
- \(1100 < N < 1200\): \(1100 < 1265 < 1200\) is false.
- \(1200 < N < 1300\): \(1200 < 1265 < 1300\) is true.
Therefore, the correct statement is \(1200 < N < 1300\).
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 \((x + y)\)?
- What is the HCF of \(2^{36}-1\) and \(2^{45}-1\)?
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 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?
