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Date November 2020 Marks available 4 Reference code 20N.3.AHL.TZ0.Hdm_3
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term Find Question number Hdm_3 Adapted from N/A

Question

Write down the remainder when 142022 is divided by 7.

[1]
a.i.

Use Fermat’s little theorem to find the remainder when 142022 is divided by 17.

[4]
a.ii.

Prove that a number in base 13 is divisible by 6 if, and only if, the sum of its digits is divisible by 6.

[4]
b.i.

The base 13 number 1y93y25 is divisible by 6. Find the possible values of the digit y.

[4]
b.ii.

Markscheme

the remainder is 0       A1


[1 mark]

a.i.

14161(mod17)  (from Fermat’s little theorem)        (A1)

142022=1416×126+6        (M1)


Note: Award M1 for a RHS exponent consistent with the correct use of Fermat’s little theorem.


142022146(mod17)  (15(mod17))       A1

the remainder is 15       A1


[4 marks]

a.ii.

METHOD 1

let N=an13n+an-113n-1++a113+a0       M1


Note: The above M1 is independent of the A marks below.


131(mod6)       A1


EITHER

13x1(mod6)  (for all x)       A1


OR

Nan1n+an-11n-1++a11+a0(mod6)  (Nan+an-1++a1+a0(mod6))       A1


THEN

so N0(mod6) if and only if an+an-1++a1+a00(mod6)       R1

so 6N if and only if 6(an+an-1++a1+a0)       AG

 

METHOD 2

let N=an13n+an-113n-1++a113+a0       (M1)

N=(an+an-1++a1+a0)+(13-1)(an(13n-1++130)+an-1(13n-2++130)++a1130)       M1A1


Note: Award M1 for attempting to express N in the form N=(an+an-1++a1+a0)+(13-1)M.


as 6(13-1)M       R1

so 6N if and only if 6(an+an-1++a1+a0)       AG


[4 marks]

b.i.

METHOD 1

the sum of the digits is 2y+20       (A1)

uses 2y+20=6k  (or equivalent) to attempt to find a valid value of y       (M1)

y=2,5,8,11(B)       A1A1


Note: Award A1 for y=2,5,8 and A1 for y=11(B).

 

METHOD 2

(1y93y25)13=1×136+y×135+9×134+3×133+y×132+2×131+5×130       (A1)

=371462y+5090480

attempts to find a valid value of y such that

371462y+50904800(mod6)       (M1)

y=2,5,8,11(B)       A1A1


Note: Award A1 for y=2,5,8 and A1 for y=11(B).


[4 marks]

b.ii.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.i.
[N/A]
b.ii.

Syllabus sections

Topic 1—Number and algebra » SL 1.7—Laws of exponents and logs
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Topic 1—Number and algebra

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