<p>If <i>7</i><sup>103</sup> is divided by 25, find the remainder.</p>
Step-by-Step Solution
Key Concept: Express the base in the form (b ± 1) where b is the divisor or related to it, then apply binomial theorem to isolate the divisor
<p><strong>Step 1:</strong> Write <i>7</i><sup>103</sup> = <i>7</i> · <i>7</i><sup>102</sup> = <i>7</i>(<i>7</i><sup>2</sup>)<sup>51</sup> = <i>7</i>(49)<sup>51</sup></p><p><strong>Step 2:</strong> Express 49 as (50 - 1):<br/>7(50 - 1)<sup>51</sup> = <i>7</i>[C<sub>0</sub>(50)<sup>51</sup> - C<sub>1</sub>(50)<sup>50</sup> + C<sub>2</sub>(50)<sup>49</sup> - ... + 1]</p><p><strong>Step 3:</strong> Expand using binomial theorem:<br/>= <i>7</i>[C<sub>0</sub>(50)<sup>51</sup> - C<sub>1</sub>(50)<sup>50</sup> + C<sub>2</sub>(50)<sup>49</sup> - ... + C<sub>50</sub>(50)]</p><p><strong>Step 4:</strong> Factor out and simplify:<br/>= <i>7</i>[50(<i>k</i>)] - 25 + 18, where <i>k</i> is an integer<br/>= 25<i>p</i> + 18, where <i>p</i> is an integer</p><p><strong>Step 5:</strong> Therefore, <i>7</i><sup>103</sup>/25 leaves remainder 18.</p><p>∴ The remainder is <strong>18</strong>.</p>
Correct Answer: 18