10 trains travelled either to or from Ingolstadt.
How many trains travelled either to or from Jena?
133. Largest possible remainder
What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits?
134. nth term n
The first term of a sequence of positive integers is 6. The other terms in the sequence follow these rules:
if a term is even then divide it by 2 to obtain the next term;
if a term is odd then multiply it by 5 and subtract 1 to obtain the next term.
For which values of n is the nth term equal to n?
135. How many numbers?
Rafael writes down a five-digit number whose digits are all distinct, and whose first digit is equal to the sum of the other four digits.
How many five-digit numbers with this property are there?
136. A square area
A regular octagon is placed inside a square, as shown.
The shaded square connects the midpoints of four sides of the octagon.
What fraction of the outer square is shaded?
137. ODD plus ODD is EVEN
Find all possible solutions to the ‘word sum’ shown.
Each letter stands for one of the digits 0−9 and has the same meaning each time it occurs. Different letters stand for different digits. No number starts with a zero.
138. Only odd digits
How many three-digit multiples of 9 consist only of odd digits?
139. How many tests?
Before the last of a series of tests, Sam calculated that a mark of 17 would enable her to average 80 over the series, but that a mark of 92 would raise her average mark over the series to 85.
How many tests were in the series?
140. Three primes
Find all positive integers p such that p, p + 8 and p + 16 are all prime.
ACROSS
1. The cube of a square (5)
4. Eight less than 5 DOWN (3)
6. One less than a multiple of seven (3)
7. A prime factor of 20 902 (4)
10. A number whose digits successively decrease by one (3)
12. Sixty per cent of 20 DOWN (3)
14. A multiple of seven (3)
15. A multiple of three whose digits have an even sum (3)
16. The square of a square (3)
17. A prime that is one less than a multiple of six (3)
19. Eleven more than a cube (4)
22. A number all of whose digits are the same (3)
24. A number that leaves a remainder of eleven when divided by thirteen (3)
25. The square of a prime; the sum of the digits of this square is ten (5)
DOWN
1. A number with an odd number of factors (3)
2. Four less than a triangular number (3)
3. The square root of 9 DOWN (2)
4. A factor of 12 ACROSS (3)
5. The longest side of a right-angled triangle whose shorter sides are 3 DOWN and 4 ACROSS (3)
8. A Fibonacci number (5)
9. The square of 3 DOWN (4)
11. Three more than an even cube (5)
13. A prime factor of 34567 (4)
17. The mean of 10 ACROSS, 16 ACROSS, 18 DOWN, 20 DOWN and 21 DOWN (3)
18. A power of eighteen (3)
20. Two less than 22 ACROSS (3)
21. A number whose digits are those of 12 ACROSS reversed (3)
23. A multiple of twenty-three (2)
141. Joey’s and Zoë’s sums
Joey calculated the sum of the largest and smallest two-digit numbers that are multiples of three. Zoë calculated the sum of the largest and smallest two-digit numbers that are not multiples of three.
What is the difference between their answers?
142. When is the party?
Six friends are having dinner together in their local restaurant. The first eats there every day, the second eats there every other day, the third eats there every third day, the fourth eats there every fourth day, the fifth every fifth day and the sixth eats there every sixth day. They agree to have a party the next time they all eat together there. In how many days’ time is the party?
143. A multiple of 11
The eight-digit number ‘1234d678’ is a multiple of 11.
Which digit is d?
144.