// Numbas version: exam_results_page_options {"name": "Modulo arithmetic: the least residue of a given number", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "functions": {}, "statement": "

The set of integers $\\{0, 1, 2,\\ldots, n-1\\}$ is called the set of least residues modulo $n$.

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One (slow!) way to determine which least residue is congruent to a number, is by repeatedly subtracting off $n$ until the result is one of the numbers from $0$ to $n-1$ (inclusive). A faster way is to divide the number by $n$ and then the remainder is the least residue.

a) We want to determine the remainder when we divide $\\var{a}$ by $2$. Since

$\\var{a}\\div 2=\\var{floor(a/2)}$$\\frac{\\var{lr_a}}{2}$

the remainder, and hence the least residue, is $\\var{lr_a}$.

Note: In the case of mod $2$ the least residue indicates whether the number is odd or even. If the number is even the least residue will be $0$, but if the number is odd the least residue will be $1$!

b) We want to determine the remainder when we divide $\\var{b}$ by $10$. Since

$\\var{b}\\div 10=\\var{floor(b/10)}$$\\frac{\\var{lr_b}}{10}$

the remainder, and hence the least residue, is $\\var{lr_b}$.

Note: In the case of mod $10$ the least residue is simply the last digit of the number!

c) We want to determine the remainder when we divide $\\var{c}$ by $100$. Since

$\\var{c}\\div 100=\\var{floor(c/100)}$$\\frac{\\var{lr_c}}{100}$

the remainder, and hence the least residue, is $\\var{lr_c}$.

Note: In the case of mod $100$ the least residue is simply the last two digits of the number!

d) We want to determine the remainder when we divide $\\var{d}$ by $\\var{small_n}$. Since

$\\var{d}\\div \\var{small_n}=\\var{floor(d/small_n)}$$\\frac{\\var{lr_d}}{\\var{small_n}}$

the remainder, and hence the least residue, is $\\var{lr_d}$.

e) We want to determine the remainder when we divide $\\var{parte}$ by $\\var{large_n}$. Since

$\\var{parte}\\div \\var{large_n}=\\var{floor(parte/large_n)}$$\\frac{\\var{lr_e}}{\\var{large_n}}$

the remainder, and hence the least residue, is $\\var{lr_e}$.

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The number $\\var{a}$ is congruent (mod $2$) to the least residue [[0]].

The number $\\var{b}$ is congruent (mod $10$) to the least residue [[0]].

The number $\\var{c}$ is congruent (mod $100$) to the least residue [[0]].

The number $\\var{d}$ is congruent (mod $\\var{small_n}$) to the least residue [[0]].

The number $\\var{parte}$ is congruent (mod $\\var{large_n}$) to the least residue [[0]].

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determine the least residue mod n of a given number.

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