CBSE Class 8 Mathematics Playing with Numbers Notes Set 11

Generalised Form of Numbers

A two-digit number ab = 10a + b and its reverse is ba = 10b + a, so ab - ba = 9(a - b) is divisible by 9.
A three-digit number abc = 100a + 10b + c; abc - cba = 99(a - c). The first digit of a number cannot be zero. Also abc + bca + cab = 111(a + b + c) = 3 x 37 x (a + b + c), so it is divisible by 37, 3 and a + b + c.

Pythagorean Triplets

Three numbers a, b, c with a² + b² = c² form a Pythagorean triplet, for example (3, 4, 5), (5, 12, 13), (6, 8, 10).
For any natural number m > 1, (2m, m² - 1, m² + 1) is a triplet. Not all triplets arise this way; (5, 12, 13) is an example.

Divisibility Tests

By 7: subtract twice the ones digit from the rest (119: 11 - 18 = -7). By 13: add four times the ones digit to the rest (611: 61 + 4 = 65). By 25: last two digits 00, 25, 50 or 75. Tests for 2, 3, 4, 5, 6, 8, 9, 11 are the usual digit rules.

Patterns

The sum of the first n odd numbers is n². The difference of a number and its reverse is a multiple of 9 (two digits) or 99 (three digits). The product of four consecutive numbers plus 1 is a perfect square: 1 x 2 x 3 x 4 + 1 = 25 = 5².

Worked Example

Find a Pythagorean triplet with 12 as a member.
Take 2m = 12, m = 6: m² - 1 = 35 and m² + 1 = 37. The triplet is 12, 35, 37.