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The number 27 is special because it is three times the sum of its digits27 = 3 (2 + 7).
Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)?
Are there any three digit 'seven-up' numbers. Is it possible to find a number K such that there is a three digit number that is K times the sum of its digits (a K-up number)?
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?
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.
Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.