AS June 2025 Q2

EdexcelCurrent spec8 marksGroups Number Theory

(i) Using a suitable algorithm and without performing any division, determine whether 13 306 617 is divisible by 9 (2)

(ii) The group \(G = \{1, 3, 7, 9, 11, 13, 17, 19\}\) has multiplication modulo 20 as its operation.

(a) Complete the following Cayley table for \(G\)

\(\times_{20}\)	1	3	7	9	11	13	17	19
1	1	3	7	9	11	13	17	19
3	3		1			19	11
7	7	1			17			13
9	9		3	1		17	13
11	11			19	1
13	13		11			9	1	7
17	17	11						3
19	19		13		9		3

(3)

(b) State the inverse of the element 7 (1)

(d) Write down a subgroup of \(G\) of order 4 (1)

Mark scheme (i)

Scheme	Marks	AO
\(1 + 3 + 3 + 6 + 6 + 1 + 7 = 27\)	M1	1.1b
The sum of the digits is 27 which is a multiple of 9 so 13 306 617 is divisible by 9	A1	1.1b
	(2)

M1: Applies the divisibility test by summing the digits.

A1: Correct solution and explanation.