Numbers
Playing with numbers
Use the divisibility tests and letter-for-digit puzzles to find missing digits.
Every two-digit number is 10a + b, where a is its tens digit and b is its units digit. Writing numbers that way turns puzzles about digits into ordinary algebra — and it explains all the divisibility tests you have been using for years without knowing why they work.
The tests, and what each one looks at
Divisible by 2: the last digit is even. By 5: the last digit is 0 or 5. By 4: the last two digits make a multiple of 4. By 8: the last three digits do. By 3: the digits add to a multiple of 3. By 9: the digits add to a multiple of 9. By 6: it passes both the 2 test and the 3 test. The first four look at the end of the number because 10, 100 and 1000 are already multiples of 2, 5, 4 and 8; the digit-sum tests work because 9, 99 and 999 are all multiples of 9.
Worked example
Why does reversing a two-digit number and subtracting always give a multiple of 9?
Write the number as 10a + b, and its reverse as 10b + a.
The digits have swapped places, so what each one is worth has swapped too.
Try it together
Now let us solve a digit puzzle together: in the sum A4 + 2A = 90, the letter A stands for one digit.
Turn each written number into tens and units, add them up as algebra, then solve for the digit.
1.A4 means 10 lots of A plus a units digit. What is that units digit?
Have a go
Have a go on your own: what digit must replace the star so that 4*2 is divisible by 9?
Hint: Add the two digits you know, and see what is needed to reach the next multiple of 9.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.