Quantitative Aptitude: Finding a Number That Leaves Specific Remainders

Quantitative aptitude is a fundamental skill in mathematics, crucial for solving various numerical problems. One interesting problem involves finding a number that leaves specific remainders when divided by a sequence of numbers. Let's explore a problem where a number leaves a remainder of (n-1) when divided by (n) for (n 10, 9, 8, ldots, 1).

Problem Statement

Find a number (x) that leaves the following remainders:

(x equiv 9 pmod{10}) (x equiv 8 pmod{9}) (x equiv 7 pmod{8}) (x equiv 6 pmod{7}) (x equiv 5 pmod{6}) (x equiv 4 pmod{5}) (x equiv 3 pmod{4}) (x equiv 2 pmod{3}) (x equiv 1 pmod{2})

Solution Approach

To find such a number, we can observe that the condition (x equiv -1 pmod{n}) for (n 10, 9, 8, ldots, 1) means that (x 1) must be divisible by all the numbers from 2 to 10.

Step 1: Express the Condition Mathematically

We can write:

[x equiv -1 pmod{10}, quad x equiv -1 pmod{9}, quad ldots, quad x equiv -1 pmod{2}]

This can be rewritten as:

[x 1 equiv 0 pmod{2, 3, 4, 5, 6, 7, 8, 9, 10}]

Therefore, (x 1) must be the least common multiple (LCM) of the numbers 2 through 10.

Step 2: Calculate the Least Common Multiple

To find the LCM, we need the prime factorization of each number:

(2 2^1) (3 3^1) (4 2^2) (5 5^1) (6 2^1 times 3^1) (7 7^1) (8 2^3) (9 3^2) (10 2^1 times 5^1)

The LCM is obtained by taking the highest powers of all primes involved:

[text{lcm}(2, 3, 4, 5, 6, 7, 8, 9, 10) 2^3 times 3^2 times 5^1 times 7^1]

Calculating step-by-step:

[8 times 9 72] [72 times 5 360] [360 times 7 2520]

Hence, the least common multiple is:

[text{lcm}(2, 3, 4, 5, 6, 7, 8, 9, 10) 2520]

Step 3: Determine the Solution

Since (x 1) is the LCM, we can express (x) as:

[x 1 2520k quad text{for some integer } k]

Thus, the number (x) is:

[x 2520k - 1]

The smallest positive solution occurs when (k 1):

[x 2520 times 1 - 1 2519]

Therefore, the number you are looking for is:

[boxed{2519}]

Additional Insights

Another way to approach the same problem involves recognizing that the entries are non-redundant. Therefore, the number (x) can be expressed as:

[x -1 pmod{2520}]

Thus, the number is: