Playing with Numbers
Generalised Form of Numbers
A two-digit number with tens digit a and units digit b is written ab = 10a + b. For example, 34 = 10 × 3 + 4. Reversing the digits gives ba = 10b + a.
Example 1: Write 34 in generalised form.
10 × 3 + 4.
Example 2: Write the reverse of 'ab' in generalised form.
ba = 10b + a.
Quick recap
- Two-digit number ab = 10a + b.
- Reversed number ba = 10b + a.
✓ Quick check
The two-digit number 'ab' equals ___ ?
ab = 10a + b.
56 = 10 × 5 + ___ ?
56 = 50 + 6.
Divisibility Tests
Using the general form, a number is divisible by 3 or 9 when its digit sum is, and by 11 when the alternating digit sum is 0 or a multiple of 11. A number is divisible by 2 when it ends in an even digit.
Example 1: Is 123 divisible by 3?
Digit sum 1 + 2 + 3 = 6, which is divisible by 3, so yes.
Example 2: When is a number divisible by 9?
When its digit sum is divisible by 9.
Quick recap
- Divisible by 3/9: check the digit sum.
- Divisible by 11: check the alternating digit sum.
✓ Quick check
Is 207 divisible by 9?
Digit sum 2 + 0 + 7 = 9, divisible by 9, so yes.
Is 1234 divisible by 2?
It ends in 4 (even), so yes.
Number Puzzles
The difference of a two-digit number and its reverse is ab − ba = 9(a − b), always a multiple of 9. Their sum is ab + ba = 11(a + b), always a multiple of 11. So 52 − 25 = 27 = 9 × 3.
Example 1: Find 52 − 25.
27, which is 9 × 3.
Example 2: What is the reverse of 41?
14.
Quick recap
- ab − ba = 9(a − b); ab + ba = 11(a + b).
- Reverse-and-subtract always gives a multiple of 9.
✓ Quick check
What is 63 − 36?
63 − 36 = 27 = 9 × 3.
What is 23 + 32?
23 + 32 = 55 = 11 × 5.
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