Let the number be \(x\).
According to the problem, when \(x\) is divided by 5, 6, and 7, the remainders are 4, 5, and 6 respectively. This can be expressed as:
- \(x \equiv 4 \pmod{5}\)
- \(x \equiv 5 \pmod{6}\)
- \(x \equiv 6 \pmod{7}\)
Notice that in each case, the remainder is one less than the divisor. This means that \(x+1\) will be divisible by 5, 6, and 7. Therefore, \(x+1\) is a common multiple of 5, 6, and 7.
To find the least common multiple (LCM) of 5, 6, and 7, we find the prime factorization of each number:
- 5 = 5
- 6 = 2 × 3
- 7 = 7
The LCM is the product of the highest powers of all prime factors present in the numbers: \(LCM(5, 6, 7) = 2 × 3 × 5 × 7 = 210\).
So, \(x + 1 = 210\). Solving for \(x\), we get \(x = 210 - 1 = 209\).
Therefore, the least number which when divided by 5, 6, and 7 leaves remainders 4, 5, and 6 is 209.
Similar Questions
The HCF of the numbers 310, 208, and 180 is:
If LCM(27, n) = 54, and HCF(27, n) = 9, then the value of n is:
The HCF of two numbers is 9 and their LCM is 252. The sum of numbers is:
- What is the difference between the HCF and LCM of 184 and 345?
- The HCF of two numbers is 9 and their LCM is 252. The sum of numbers is:
The HCF of two numbers is 62, and the other two factors of the LCM of these factors are 7 and 9. Which of the options given below can be a possible value of the smaller of the two numbers?
- Three bells ring at intervals of 15, 30 and 45 minutes respectively. At what time will they ring together again, if they rang simultaneously at 8.00 AM?
The LCM of 24, 42, and 56 is:
The HCF of 450, 1050, and 1650 is:
The LCM of 84, 105, and 140 is:
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?
