Test: Playing with Numbers - Class 8 MCQ

The sum of a two-digit number 'ab' and its reverse 'ba' is always divisible by 11. This is derived from the simplification of the equation (10a + b) + (10b + a) = 11(a + b).

The number of permutations of a three-digit number can be calculated as 3! (3 factorial), which equals 6. The permutations for 437 are 437, 473, 347, 374, 743, and 734.

The arrangement 'bca' in a three-digit number involves the hundreds digit being 'b', the tens digit being 'c', and the units digit being 'a', which translates to the value of 100b + 10c + a.

The sum of the digits in 43242876 is calculated as follows: 4 + 3 + 2 + 4 + 2 + 8 + 7 + 6 = 36. This sum is also used to check divisibility by 9.

The number 300 can be expressed in its generalized form as 3 × 100 + 0 × 10 + 0 × 1, clearly indicating the value of each digit based on its place value.

The factors of a number are those integers that divide it without leaving a remainder. For the number 12, the factors are 1, 2, 3, 4, 6, and 12.

Expressing a number in its generalized form involves writing it as a sum of each digit multiplied by its respective place value, which illustrates the contribution of each digit to the overall value of the number.

A number is divisible by 4 if its last two digits form a number that is divisible by 4. Since 16 is divisible by 4, any number ending in 16 is also divisible by 4.

A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is divisible by 11. This method helps to determine divisibility effectively.

The absolute difference between a two-digit number and its reverse is always divisible by 9. This arises from the structure of how the digits are arranged and their respective values.

Factors of a number are integers that can divide that number evenly, while multiples are the products obtained when a number is multiplied by integers. This relationship is fundamental in number theory.

A number is divisible by 6 if it is simultaneously divisible by both 2 and 3. For 540, it is even (divisible by 2) and the sum of its digits (5 + 4 + 0 = 9) is divisible by 3, confirming its divisibility by 6.

In a cryptarithm, each letter represents a unique digit ranging from 0 to 9. This uniqueness is crucial for solving the puzzle accurately and ensuring that the mathematical operations hold true.

A number is divisible by 2 if its last digit (units digit) is even. This is a straightforward rule that applies to any integer.

Natural numbers are defined as counting numbers that start from 1 and include all positive integers (1, 2, 3, ...). They do not include zero or any negative numbers.