Understanding the Problem
The question asks for the smallest 4-digit number that leaves a remainder of 3 when divided by 4, 6, 7, and 9. This means the number, let's call it N, satisfies these conditions:
- \(N \div 4\) leaves a remainder of 3
- \(N \div 6\) leaves a remainder of 3
- \(N \div 7\) leaves a remainder of 3
- \(N \div 9\) leaves a remainder of 3
- \(N\) must be the smallest possible number that is greater than or equal to 1000 (the smallest 4-digit number).
Finding the Key Concept: LCM
When a number leaves the same remainder upon division by several different numbers, it suggests using the concept of the Least Common Multiple (LCM). If we subtract the common remainder (3 in this case) from the number N, the result (\(N-3\)) must be perfectly divisible by 4, 6, 7, and 9.
Therefore, \(N-3\) must be a multiple of the LCM of 4, 6, 7, and 9. The number N can be expressed as:
\(N = \text{LCM}(4, 6, 7, 9) \times k + 3\), where \(k\) is a positive integer.
Calculating the LCM of 4, 6, 7, and 9
To find the LCM, we first find the prime factorization of each number:
- \(4 = 2^2\)
- \(6 = 2 \times 3\)
- \(7 = 7\) (7 is a prime number)
- \(9 = 3^2\)
The LCM is found by taking the highest power of each prime factor present in any of the numbers:
LCM\((4, 6, 7, 9) = 2^2 \times 3^2 \times 7^1\)
LCM\((4, 6, 7, 9) = 4 \times 9 \times 7\)
LCM\((4, 6, 7, 9) = 36 \times 7\)
LCM\((4, 6, 7, 9) = 252\)
Finding the Smallest 4-Digit Number
Now we know that the number N must be in the form \(N = 252 \times k + 3\). We need the smallest such number that is a 4-digit number, meaning \(N \ge 1000\).
We can set up the inequality:
\(252 \times k + 3 \ge 1000\)
Subtract 3 from both sides:
\(252 \times k \ge 997\)
Divide by 252:
\(k \ge \frac{997}{252}\)
\(k \ge 3.956...\)
Since \(k\) must be an integer, the smallest integer value for \(k\) that satisfies this condition is \(k = 4\).
Now, substitute \(k=4\) back into the formula for N:
\(N = 252 \times 4 + 3\)
\(N = 1008 + 3\)
\(N = 1011\)
Verifying the Result
Let's check if 1011 satisfies the conditions:
- \(1011 \div 4 = 252\) with a remainder of \(1011 - (252 \times 4) = 1011 - 1008 = 3\).
- \(1011 \div 6 = 168\) with a remainder of \(1011 - (168 \times 6) = 1011 - 1008 = 3\).
- \(1011 \div 7 = 144\) with a remainder of \(1011 - (144 \times 7) = 1011 - 1008 = 3\).
- \(1011 \div 9 = 112\) with a remainder of \(1011 - (112 \times 9) = 1011 - 1008 = 3\).
The number 1011 is indeed the smallest 4-digit number that leaves a remainder of 3 when divided by 4, 6, 7, or 9.
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 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?
