You may also like

problem icon

14 Divisors

What is the smallest number with exactly 14 divisors?

problem icon

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.

problem icon

Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Weekly Problem 37 - 2006

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2
A test for divisibility by 11 is to add alternate digits:

1 + 3 + * + 7 = 11 + *; 2 + 4 + 6 + 8 = 20.

If the original number is a multiple of 11 then these two totals will be the same or will differ by a multiple of 11. In this case, 11 + * = 20 gives * = 9.

Or, you can solve it without knowing a rule as follows:

1234*678 = 12340678 + 1000* = (11 x 1121879 +9) + 11 x 90* + 10*

and hence is divisible by 11 if and only if 10* + 9 is divisible by 11. So * = 9.

This problem is taken from the UKMT Mathematical Challenges.

View the previous week's solution
View the current weekly problem
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
Copyright © 1997 - 2012. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project.