### Lastly - Well

What are the last two digits of 2^(2^2003)?

### Counting Factors

Is there an efficient way to work out how many factors a large number has?

### Summing Consecutive Numbers

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

# Alphabet Soup

##### Age 11 to 14 Challenge Level:

Keep sending us YOUR OWN alphanumerics and we'll publish them in collections from time to time. The following two came from Jonathan Gill, St Peter's College, Adelaide, Australia.

 C A + R A V A N
 A U S + T R A L I A N

There is a one-to-one correspondence between digits and letters, each letter stands for a single digit and each digit is represented by a single letter. How many different solutions can you find?

Ling Xiang Ning(Allan) form Tao Nan School, Singapore, who solves many of are hardest problems, has sent 7 solutions to CARAVAN and 88 solutions to AUSTRALIAN. Is this all there are? Here is one solution to each.

 76 968 +86 +529 ---- ---- 162 1497

Soh Yong Sheng, age 12, also from Tao Nan School, Singapore has sent this solution for.

 NRICH + STARS = MATHS 17230 + 48574 = 65804

and there are al lot more.

We have the following solutions from Allan Ling (Tao Nan School, Singapore): For the equation

 M A T H + E M A T I C A L

T has to be 9 or 0, in order for it to satisfy T+A=A. However if T=0, it is impossible, as H+0 is not L. So T has to be 9.

The following are the possible sums (total 59):

 4891 5791 4791 2591 3491 2491 2491 +2489 +2579 +3479 +4259 +2349 +6249 +3249 7380 8370 8270 6850 5840 8740 5740 4391 2391 3291 3291 3692 4592 5092 +2439 +5239 +5329 +4329 +4369 +3459 +3509 6830 7630 8620 7620 8061 8051 8601 4092 3092 5893 1893 2793 1893 2793 +3409 +5309 +1589 +4189 +5279 +5189 +1279 7501 8401 7482 6082 8072 7082 4072 1693 4593 5493 2493 1493 6093 1894 +4169 +1459 +1549 +6249 +7149 +2609 +5189 5862 6052 7042 8742 8642 8702 7083 3794 2794 6594 5294 1094 1094 1094 +2379 +5279 +1659 +1529 +7109 +6109 +5109 6173 8073 8253 6823 8203 7203 6203 4795 4795 3795 2795 1695 1095 1095 +3479 +1479 +2379 +3279 +2169 +7109 +6109 8274 6274 6174 6074 3864 8204 7204 1896 2496 1296 1096 1096 2197 1097 +2189 +1249 +7129 +7109 +3109 +3219 +4109 4085 3745 8425 8205 4205 5416 5206 1097 1498 3298 1298 2198 4098 3098 +3109 +2149 +1329 +5129 +3219 +2409 +2309 4206 3647 4627 6427 5417 6507 5407 2098 1098 1098 1098 +4209 +5109 +4109 +3109 6307 6207 5207 4207

Jonathan also proved that the following alphanumeric does not work, that is it cannot have any solutions. Well done Jonathan.

 N R I C H + M A T H S S T A R S

If it was an alphanumerics then H = 0 to satisfy 0 + S = S, but then H cannot be zero, otherwise C + 0 (H) = C and not R. We know that C and R cannot both represent the same number therefore

 N R I C H + M A T H S S T A R S

cannot be made into an alphanumeric.