Number Play

Overview

This chapter is about playing with numbers: factors, multiples, divisibility tests, and number puzzles based on place value. It builds reasoning skills and reveals hidden patterns in arithmetic.

Key concepts

• A two-digit number with digits a and b has value 10a + b.
• Divisibility tests give quick checks for 2, 3, 4, 5, 6, 8, 9, 10 and 11.
• HCF is the largest common factor; LCM is the smallest common multiple.
• Reversing digits and adding or subtracting reveals neat patterns (multiples of 9 or 11).

Important formulae

• HCF × LCM = product of the two numbers.
• Divisible by 3 or 9 if the digit sum is divisible by 3 or 9.
• Divisible by 11 if the difference of alternate digit sums is 0 or a multiple of 11.

Solved example

1. A two-digit number is 63. Reverse it to get 36 and subtract.
2. 63 = 10(6) + 3 and 36 = 10(3) + 6.
3. Difference = 9(6 − 3) = 9 × 3 = 27, a multiple of 9 as expected.

Important questions

1. Find the HCF and LCM of 24 and 36, then verify HCF × LCM = 24 × 36.
2. Check whether 91839 is divisible by 11.
3. If 31z5 is divisible by 9, find the digit z.
4. Prove that the sum of a two-digit number and its reverse is always divisible by 11.

Quick revision

Express numbers using place value to explain digit puzzles. Master the divisibility tests and the HCF × LCM relation. Reversal patterns always link back to 9 and 11.