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Adding All Nine

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

Double Digit

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?

Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Ones Only

Age 11 to 14 Challenge Level:

Lots of people found the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones. The answer is 15873 since 7 x 15873 = 111111.

The people who explained how they got this answer by working systematically using division were Martin, Richard, David and James of Hethersett High School, Norwich; Chris and Adrian of Poltair Community School, St Austell, Cornwall; Prav and Sheli from the North London Collegiate School Maths Puzzle Club; Maren and Corinna of Camden School for Girls and Soh from Raffles Institution, Singapore..

The best account was from Amy of Madras College, St Andrews. Well done Amy! She first defined a Repunit as a number consisting entirely of 1's so that R 1 = 1; R 2 = 11; R 3 = 111; R 4 = 1111 etc.

Investigating by division:

11 / 7 = 1.571...
111 / 7 = 15.857...
1111 / 7 = 158.714...
11111 / 7 = 11587.28...
111111 / 7 = 15873.

So 15873 is the smallest whole number which, when multiplied by 7, gives a repunit.

Extending the question to find what other numbers will also work, we know 3 must be the smallest number to work because a number ending in a 1 will be odd so it will not divide by 2.

11 / 3 = 3.666...
111 / 3 = 37 .

So 37 is the smallest whole number which, when multiplied by 3, gives a repunit.

The number 4 will not work for the same reason that 2 does not work, and 6 won't work, neither will 8.. Any number multiplied by 5 will end in a 0 or 5, so 5 won't work, so 9 will be the next number to try.

11 / 9 = 1.222...
R 3 / 9 = 12. 33...
R 4 / 9 = 123.44...
R 5 /9 = 1234.55...
R 6 / 9 = 12345.66...
R 7 / 9 = 123456.77...
R 8 / 9 = 1234567.88...
R 9 /9 = 123456789

So 123456789 is the smallest whole number which, when multiplies by 9, gives a repunit.