Simplifying with Identities and Approximation

Spotting the Identity

A lot of SSC simplification questions are one algebra identity with real numbers put in. Once you see the identity, a page of multiplication turns into one line. Here are the ones you need. In this section $a$, $b$ and $c$ are always plain numbers, like 4.7 or 997. The same identities with letters, and the whole $x + \frac{1}{x}$ family, are in the Algebra chapter.

How SSC hides them.

• Squares and cubes are written as products: 4.7 × 4.7 instead of $4.7^2$, or 0.6 × 0.6 × 0.6 instead of $0.6^3$.
• The $2ab$ or $3ab$ term is written out as "2 × 4.7 × 5.3", or it is already multiplied out into a single number.
• Squares and cubes of decimals are given as values: 0.49 instead of $0.7^2$, 0.343 instead of $0.7^3$.
• The numbers are picked so that $a + b$ or $a - b$ is a round number like 1, 10 or 100.

That last point is your signal. If two numbers in a question add up to, or differ by, something round, an identity is almost certainly behind it.

For example, take 4.7 × 4.7 + 2 × 4.7 × 5.3 + 5.3 × 5.3. Name $a = 4.7$ and $b = 5.3$. The expression is $a^2 + 2ab + b^2 = (a + b)^2$, and 4.7 + 5.3 = 10, so the answer is 100. Doing the three multiplications instead would take a minute and give the same 100.

Squares: Full Squares and the Sum-Difference Pair

Full squares in disguise. $a^2 + 2ab + b^2$ folds up into $(a + b)^2$, and $a^2 - 2ab + b^2$ folds up into $(a - b)^2$.

$$6.3^2 - 2 \times 6.3 \times 1.3 + 1.3^2 = (6.3 - 1.3)^2 = 5^2 = 25$$

Sometimes the middle term is already multiplied out, as in $2.5^2 + 1.5^2 + 7.5$. Ask whether 7.5 is $2 \times 2.5 \times 1.5$. It is, so the answer is $(2.5 + 1.5)^2 = 16$.

Finding the missing term. "What should replace ? to make $3.2^2$ + ? + $1.8^2$ a perfect square?" The middle term must be $2ab$ = 2 × 3.2 × 1.8 = 11.52, and then the whole thing is $(3.2 + 1.8)^2 = 25$.

Squaring a number near a round number. Split it as round number ± small part:

• $1004^2 = (1000 + 4)^2$ = 10,00,000 + 8,000 + 16 = 10,08,016
• $9.98^2 = (10 - 0.02)^2$ = 100 − 0.4 + 0.0004 = 99.6004

The Squares, Square Roots and Cube Roots section has more squaring tricks. Here the point is to recognise the identity inside a longer expression.

The sum-difference pair. Identities 4 and 5 put $(a + b)^2$ and $(a - b)^2$ side by side:

$$\frac{(a + b)^2 + (a - b)^2}{a^2 + b^2} = 2 \qquad \frac{(a + b)^2 - (a - b)^2}{ab} = 4$$

So $\frac{(376 + 124)^2 - (376 - 124)^2}{376 \times 124} = 4$ without a single multiplication, and $\frac{(0.9 + 0.4)^2 + (0.9 - 0.4)^2}{0.9^2 + 0.4^2} = 2$.

Watch the bottom of the fraction. The "+" version goes with $a^2 + b^2$ and gives 2. The "−" version goes with $ab$ and gives 4. If the bottom is $2(a^2 + b^2)$ or $4ab$ instead, the answer is 1.

Two more forms of the same idea.

$$(a + b)^2 - 4ab = (a - b)^2 \qquad (a - b)^2 + 4ab = (a + b)^2$$

So $(5.6 + 4.4)^2 - 4 \times 5.6 \times 4.4 = (5.6 - 4.4)^2 = 1.2^2 = 1.44$.

Difference of Squares

$a^2 - b^2 = (a + b)(a - b)$ comes up often in calculation questions. It works in both directions.

Direction 1: subtracting two squares. Don't square anything. Add and subtract the numbers instead.

$$67^2 - 33^2 = (67 + 33)(67 - 33) = 100 \times 34 = 3400 \qquad 8.5^2 - 1.5^2 = 10 \times 7 = 70$$

When the question then divides by $a - b$ or by $a + b$, it is handing you the answer:

$$\frac{a^2 - b^2}{a - b} = a + b \qquad \frac{a^2 - b^2}{a + b} = a - b$$

So $\frac{8.6^2 - 3.4^2}{5.2}$ = 8.6 + 3.4 = 12, because 8.6 − 3.4 is exactly 5.2.

Direction 2: multiplying two numbers the same distance from a round number. Write them as $(m + d)(m - d) = m^2 - d^2$, where $m$ is the middle and $d$ is the distance.

The middle doesn't have to be 100 or 1000. Anything with an easy square works: 32 × 28 = 900 − 4 = 896.

Chains of brackets. $(a - b)(a + b) = a^2 - b^2$, then $(a^2 - b^2)(a^2 + b^2) = a^4 - b^4$, and so on. So a product like $(2 + 1)(2^2 + 1)(2^4 + 1)$ becomes easy if you multiply it by $(2 - 1)$, which is just 1:

$$(2 - 1)(2 + 1)(2^2 + 1)(2^4 + 1) = (2^4 - 1)(2^4 + 1) = 2^8 - 1 = 255$$

If the base isn't 2, the extra bracket isn't 1. For base 3 you multiply by $(3 - 1) = 2$ to fold the chain, so you must halve at the end. For base 5, divide by 4.

[SSC Tip] Two numbers that add up to a round number and appear squared are a strong hint for $a^2 - b^2$. Two numbers that are the same distance from a round number and appear multiplied are the same hint the other way round.

Multiplying Numbers Near a Base

When both numbers are close to the same power of 10 but not the same distance from it, use the base method. It is just $(B + x)(B + y) = B(B + x + y) + xy$ written out in a quick form.

1. Write how far each number is from the base: + if above, − if below.
2. Left part: one number plus the other number's distance. (Both ways give the same.)
3. Right part: multiply the two distances. It must have as many digits as the base has zeros.
4. Put the two parts side by side.

Padding and carrying. With base 1000 the right part needs three digits, so 44 becomes 044. If the right part has too many digits, carry the extra to the left. For 88 × 87 the distances are −12 and −13, so the left part is 75 and the right part is 156. Base 100 allows only two digits, so the 1 carries: 75 + 1 = 76, giving 7,656.

One above, one below. The right part comes out negative, so you subtract it. For 104 × 97 the distances are +4 and −3. The left part is 104 − 3 = 101, which stands for 10,100, and the right part is −12. So the answer is 10,100 − 12 = 10,088.

Decimals. Ignore the point, multiply, then put the point back. For 9.8 × 9.7, work out 98 × 97: left 95, right 06, so 9506. There are two decimal places in all, so the answer is 95.06.

When the two distances are equal and opposite, as in 98 × 102, the base method gives the same thing as $a^2 - b^2$: 10,000 − 4 = 9,996.

The base method is handy in "what comes in place of ?" questions where a product near 100 or 1000 sits inside a longer expression. Check the result with the last digit: 96 × 89 must end in 4, because 6 × 9 = 54.

Cube Identities with Decimals

The full cube. $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$. SSC likes numbers with $a + b = 1$, because then the $(a + b)$ quietly drops out of the last term:

$$0.62^3 + 0.38^3 + 3 \times 0.62 \times 0.38 = (0.62 + 0.38)^3 = 1$$

If $a + b = 2$, the last term must carry the extra × 2: $1.3^3 + 0.7^3 + 3 \times 1.3 \times 0.7 \times 2 = 2^3 = 8$. If that × 2 is missing, it is not a full cube, so read the term carefully.

The minus version is $a^3 - b^3 - 3ab(a - b) = (a - b)^3$. So $1.6^3 - 0.6^3 - 3 \times 1.6 \times 0.6 = 1^3 = 1$, because here $a - b = 1$.

Sum or difference of cubes over the matching bracket.

$$\frac{a^3 + b^3}{a^2 - ab + b^2} = a + b \qquad \frac{a^3 - b^3}{a^2 + ab + b^2} = a - b$$

$$\frac{7.3^3 + 2.7^3}{7.3^2 - 7.3 \times 2.7 + 2.7^2} = 7.3 + 2.7 = 10 \qquad \frac{2.4^3 - 1.4^3}{2.4^2 + 2.4 \times 1.4 + 1.4^2} = 2.4 - 1.4 = 1$$

Remember how the signs pair up: the plus in $a^3 + b^3$ goes with the minus in $a^2 - ab + b^2$, and the other way round. If the bracket is the other one, there is no shortcut.

Cubes and squares given as values. Learn the cubes of 0.1 to 0.9 so you can spot them inside a question:

So $\frac{0.343 + 0.125}{0.49 - 0.35 + 0.25}$ is $\frac{a^3 + b^3}{a^2 - ab + b^2}$ with $a = 0.7$ and $b = 0.5$ (0.35 is 0.7 × 0.5). The answer is 0.7 + 0.5 = 1.2.

Scaled ratios. If every number on top is the same multiple $k$ of the matching number below, a ratio of cube sums is $k^3$ (and a ratio of square sums is $k^2$):

$$\frac{0.24^3 + 0.08^3}{0.6^3 + 0.2^3} = 0.4^3 = 0.064$$

This works because 0.24 = 0.4 × 0.6 and 0.08 = 0.4 × 0.2, so the top is $0.4^3 \times (0.6^3 + 0.2^3)$.

[Tier 2] A fourth-power cousin. $a^4 + a^2 b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)$. So $\frac{3^4 + 3^2 \times 2^2 + 2^4}{3^2 + 3 \times 2 + 2^2} = 3^2 - 3 \times 2 + 2^2 = 7$. Check: 133 ÷ 19 = 7.

Three Numbers: $a^3 + b^3 + c^3 - 3abc$

$$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$$

The second bracket has a friendlier form that is easier to work out:

$$a^2 + b^2 + c^2 - ab - bc - ca = \frac{1}{2}\left[(a - b)^2 + (b - c)^2 + (c - a)^2\right]$$

Case 1: the three numbers add to 0. Then the whole right side is 0, so

$$a^3 + b^3 + c^3 = 3abc$$

Look for three numbers where one is the negative of the other two added together.

$$\frac{0.5^3 + 0.4^3 - 0.9^3}{3 \times 0.5 \times 0.4 \times 0.9}$$

Take $a = 0.5$, $b = 0.4$ and $c = -0.9$. They add to 0, so the top is $3abc = 3 \times 0.5 \times 0.4 \times (-0.9)$. The bottom is the same without the minus, so the answer is $-1$. Notice two things. The minus sign comes from $c$ being negative. And the 3 matters: if the bottom were only 0.5 × 0.4 × 0.9, the answer would be $-3$.

The same idea handles differences, because $(x - y) + (y - z) + (z - x)$ is always 0:

$$(x - y)^3 + (y - z)^3 + (z - x)^3 = 3(x - y)(y - z)(z - x)$$

For $x = 5$, $y = 2$ and $z = 9$: $3^3 + (-7)^3 + 4^3$ = 27 − 343 + 64 = −252, and 3 × 3 × (−7) × 4 = −252 as well.

Case 2: three numbers close together. Use the half-sum-of-squares form. For 41, 42 and 43 the differences are 1, 1 and 2, so the bracket is half of (1 + 1 + 4), which is 3. The sum of the numbers is 126, so $41^3 + 42^3 + 43^3 - 3 \times 41 \times 42 \times 43$ = 126 × 3 = 378.

Case 3: the long bracket is at the bottom.

$$\frac{a^3 + b^3 + c^3 - 3abc}{a^2 + b^2 + c^2 - ab - bc - ca} = a + b + c$$

Case 4: all three equal. If $a = b = c$, the expression is 0.

The square of three numbers. $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$. So $2.3^2 + 1.4^2 + 0.3^2 + 2(2.3 \times 1.4 + 1.4 \times 0.3 + 0.3 \times 2.3)$ is just $(2.3 + 1.4 + 0.3)^2 = 4^2 = 16$.

Approximation: "The Value Is Closest To"

These questions say "approximately" or "closest to" because the exact answer is ugly. The options are usually far enough apart that rounding gets you there, so working out the exact value is wasted time.

Step 1. Look at the options first. If they are 380, 400, 420 and 440, you need to land within about 10. If they are 200, 400, 600 and 800, rough rounding is plenty.

Step 2. Round every number to something friendly.

• Decimals to the nearest whole or round number: 39.97 → 40, 4.98 → 5, 124.98 → 125.
• Percentages to easy fractions: 19.97% → 20% = $\frac{1}{5}$, 33.4% → $\frac{1}{3}$, 12.4% → $\frac{1}{8}$.
• Roots to the nearest perfect square: $\sqrt{168} \approx \sqrt{169} = 13$, $\sqrt{401} \approx 20$.
• Awkward fractions to decimals you know: $\frac{1}{7} \approx 0.143$, $\frac{2}{3} \approx 0.667$.

Step 3. Notice which way you rounded. In a product, rounding one number up and another down partly cancels out. In a division, rounding the top and the bottom the same way partly cancels. If you rounded everything up in a product, expect the true answer to be a little below yours.

Example: $39.97 \times 5.02 + 15.03^2 - 124.98 \div 4.99$. Rounded, it is 40 × 5 + $15^2$ − 125 ÷ 5 = 200 + 225 − 25 = 400. The exact value is about 401.5, so 400 is the closest option.

Small-change facts. For a small number $x$ (like 0.01 or 0.02):

• $(1 + x)^2 \approx 1 + 2x$, so $1.02^2 \approx 1.04$. (Exactly, it is 1.0404.)
• $\frac{1}{1 - x} \approx 1 + x$, so $\frac{1}{0.98} \approx 1.02$.
• $\sqrt{n^2 + d} \approx n + \frac{d}{2n}$ when $d$ is small, so $\sqrt{50} \approx 7 + \frac{1}{14} \approx 7.07$. Rough roots are covered in more detail in the Squares, Square Roots and Cube Roots section.

Step 4. Use the identities here too. 49.9 × 50.1 is $50^2 - 0.1^2$, so it is a hair under 2500. An approximation question often hides an exact shortcut.

[SSC Tip] If two options are close, say 398 and 412, round less. Keep one decimal place in the biggest term, because that term carries most of the error. Rounding 15.03 to 15 changes $15.03^2$ by almost 1, while rounding 4.99 to 5 in a divisor hardly matters.

Checking Answers with Digit Sums

Digit sums can clear options in a product without doing it. In simplification questions the same check works at almost every step, and it costs a few seconds.

The digit sum. Add the digits, and keep adding until one digit is left. Treat 9 as 0. So 7,846 → 25 → 7. A number and its digit sum leave the same remainder when divided by 9, which is why the rules below work.

The rules.

• Addition: the digit sum of the answer equals the digit sums added (and reduced).
• Subtraction: the same with subtraction. If you get a negative number, add 9.
• Multiplication: multiply the digit sums, then reduce.
• Squares and cubes: square or cube the digit sum, then reduce.
• Division: don't divide digit sums. Check instead that divisor × quotient (+ remainder) gives the dividend.

Decimals. Moving the decimal point only multiplies by a power of 10, and that doesn't change the digit sum. So 3.86 × 2.7 can be checked as 386 × 27. You still have to count decimal places separately to put the point in the right place.

Example. Is 738 × 46 equal to 33,848 or 33,948? Both end in 8, so the last digit is no help. 738 has digit sum 18 → 9, which counts as 0, so the product must have digit sum 9 too. 33,948 → 27 → 9 works and 33,848 → 26 → 8 doesn't. So the answer is 33,948.

Using it on a whole expression. Say you worked out 12,000 − 457 × 23 and got 1,489. Check it: 12,000 has digit sum 3. 457 → 7 and 23 → 5, so 457 × 23 → 7 × 5 = 35 → 8. Then 3 − 8 = −5, and adding 9 gives 4. 1,489 → 22 → 4 matches, so the answer is very likely right.

What digit sums can't catch.

• Two digits swapped: 33,948 and 39,348 have the same digit sum.
• A missing or extra 0 or 9.
• A decimal point in the wrong place.

When two options share a digit sum, use the last digit or a rough size check instead. Because the digit sum can only rule options out, use it as a final check, not as your first method.

Solved Examples

Question 1

Find the value of $3.68 \times 3.68 + 2 \times 3.68 \times 1.32 + 1.32 \times 1.32$.

Answer. With $a = 3.68$ and $b = 1.32$, this is $a^2 + 2ab + b^2 = (a + b)^2$. And 3.68 + 1.32 = 5, so the value is $5^2$ = 25.

Question 2

Find $997^2$ without long multiplication.

Answer. 997 = 1000 − 3, so I use $(a - b)^2 = a^2 - 2ab + b^2$. That gives 10,00,000 − 2 × 1000 × 3 + 9 = 10,00,000 − 6,000 + 9 = 9,94,009.

Question 3

Find $\frac{(0.87 + 0.13)^2 + (0.87 - 0.13)^2}{0.87^2 + 0.13^2}$.

Answer. The top is $(a + b)^2 + (a - b)^2$, which is always $2(a^2 + b^2)$, and the bottom is $a^2 + b^2$. So the answer is 2, whatever $a$ and $b$ are. If you want to see it with numbers: the top is 1 + 0.5476 = 1.5476, the bottom is 0.7569 + 0.0169 = 0.7738, and 1.5476 ÷ 0.7738 = 2.

Question 4

Find 98.5 × 101.5.

Answer. Both numbers are 1.5 away from 100, one below and one above. So the product is $(100 - 1.5)(100 + 1.5) = 100^2 - 1.5^2$ = 10,000 − 2.25 = 9,997.75.

Question 5

Find $\frac{7.65^2 - 2.35^2}{5.3}$.

Answer. I notice that 7.65 − 2.35 = 5.3, which is exactly the bottom. Since $a^2 - b^2 = (a + b)(a - b)$, dividing by $a - b$ leaves $a + b$ = 7.65 + 2.35 = 10.

Question 6

Find 97 × 94 and 1006 × 1013 by the base method.

Answer. For 97 × 94 the base is 100 and the distances are −3 and −6. Left part: 97 − 6 = 91. Right part: (−3) × (−6) = 18, which fits in two digits. So 97 × 94 = 9,118.

For 1006 × 1013 the base is 1000 and the distances are +6 and +13. Left part: 1006 + 13 = 1019. Right part: 6 × 13 = 78, which I write with three digits as 078. So 1006 × 1013 = 10,19,078.

Question 7

Find $\frac{6.8^3 + 3.2^3}{6.8^2 - 6.8 \times 3.2 + 3.2^2}$.

Answer. The top is $a^3 + b^3$ and the bottom is $a^2 - ab + b^2$, with $a = 6.8$ and $b = 3.2$. Since $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, the fraction is just $a + b$ = 6.8 + 3.2 = 10.

Question 8

Find $\frac{0.85^3 - 0.35^3}{0.85^2 + 0.85 \times 0.35 + 0.35^2}$.

Answer. This time it is $a^3 - b^3$ over $a^2 + ab + b^2$. The minus on top goes with the plus in the bracket, so the pairing is right, and the fraction is $a - b$ = 0.85 − 0.35 = 0.5.

Question 9

Find $0.64^3 + 0.36^3 + 3 \times 0.64 \times 0.36$.

Answer. 0.64 + 0.36 = 1, and that's the signal. The full cube is $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$, and when $a + b = 1$ the last term is just $3ab$. So the expression is $(0.64 + 0.36)^3 = 1^3$ = 1.

Question 10

Find $\frac{0.08^3 + 0.06^3}{0.4^3 + 0.3^3}$.

Answer. I compare the numbers on top with the ones below: 0.08 = 0.2 × 0.4 and 0.06 = 0.2 × 0.3. So each number on top is 0.2 times the matching number below. Cubing, each cube on top is $0.2^3$ times the one below, so the whole fraction is $0.2^3$ = 0.008.

Question 11

Find $\frac{0.512 - 0.027}{0.64 + 0.24 + 0.09}$.

Answer. I look for cubes and squares I know. 0.512 is $0.8^3$ and 0.027 is $0.3^3$. On the bottom, 0.64 is $0.8^2$, 0.09 is $0.3^2$, and 0.24 is 0.8 × 0.3. So it is $\frac{a^3 - b^3}{a^2 + ab + b^2}$ with $a = 0.8$ and $b = 0.3$, which equals $a - b$ = 0.5.

Question 12

Find $\frac{0.5^3 + 0.2^3 - 0.7^3}{0.5 \times 0.2 \times 0.7}$.

Answer. I take $a = 0.5$, $b = 0.2$ and $c = -0.7$. They add to 0, so $a^3 + b^3 + c^3 = 3abc$, and the top is $3 \times 0.5 \times 0.2 \times (-0.7)$. The bottom has no 3 and no minus sign, so the answer is $-3$.

Question 13

Find $101^3 + 102^3 + 103^3 - 3 \times 101 \times 102 \times 103$.

Answer. This is $a^3 + b^3 + c^3 - 3abc$, which equals (the sum of the numbers) × (half the sum of the squared differences). The differences are 1, 1 and 2, so the second part is half of (1 + 1 + 4) = 3. The sum is 101 + 102 + 103 = 306. So the value is 306 × 3 = 918.

Question 14

Find $(47 - 32)^3 + (32 - 19)^3 + (19 - 47)^3$.

Answer. The three brackets are 15, 13 and −28, and they add to 0, as they always do in this pattern. So the sum of their cubes is 3 times their product: 3 × 15 × 13 × (−28) = −16,380.

Question 15

Find $(3.79 + 2.21)^2 - 4 \times 3.79 \times 2.21$.

Answer. $(a + b)^2 - 4ab = (a - b)^2$. Here $a - b$ = 3.79 − 2.21 = 1.58, so the answer is $1.58^2$. I square it as $(1.6 - 0.02)^2$ = 2.56 − 0.064 + 0.0004 = 2.4964.

Question 16

Find $(3 + 1)(3^2 + 1)(3^4 + 1)$.

Answer. I multiply by $(3 - 1)$ so the brackets fold up one after another: $(3 - 1)(3 + 1) = 3^2 - 1$, then $(3^2 - 1)(3^2 + 1) = 3^4 - 1$, then $(3^4 - 1)(3^4 + 1) = 3^8 - 1$ = 6560. But I multiplied by 2 to get there, so I halve at the end: 6560 ÷ 2 = 3280. A direct check agrees: 4 × 10 × 82 = 3280.

Question 17

Which is closest to 29.97 × 40.02 ÷ 5.99? (a) 180 (b) 200 (c) 220 (d) 240

Answer. Rounding, it is 30 × 40 ÷ 6 = 1200 ÷ 6 = 200. The options are 20 apart, so this rounding is easily close enough. The answer is (b) 200.

Question 18

Find the approximate value of $\sqrt{325} \times 4.98$ + 19.8% of 501. (a) 190 (b) 210 (c) 230 (d) 250

Answer. $\sqrt{325}$ is just above $\sqrt{324} = 18$, so the first part is about 18 × 5 = 90. And 19.8% of 501 is about 20% of 500, which is 100. The total is about 190, so the answer is (a) 190. (The exact value is about 189.)

Question 19

Which is closest to $15.02^2 + 7.98 \times 12.03 - \sqrt{399}$? (a) 280 (b) 290 (c) 300 (d) 310

Answer. $15^2$ = 225, 8 × 12 = 96, and $\sqrt{399}$ is just under $\sqrt{400} = 20$. So it is about 225 + 96 − 20 = 301. The options are only 10 apart, but 301 is clearly nearest to (c) 300.

Question 20

4,876 × 239 = ? (a) 11,65,264 (b) 11,65,364 (c) 11,65,354 (d) 11,66,364

Answer. All four end in 4 (6 × 9 = 54), so the last digit doesn't help, and I use digit sums. 4876 → 4 + 8 + 7 + 6 = 25 → 7, and 239 → 14 → 5. So the product must have digit sum 7 × 5 = 35 → 8. The options give (a) 25 → 7, (b) 26 → 8, (c) 25 → 7 and (d) 27 → 9. Only (b) 11,65,364 has digit sum 8.

Question 21

8,46,264 ÷ 74 = ? (a) 11,446 (b) 11,426 (c) 11,436 (d) 11,336

Answer. All four end in 6, so the last digit can't separate them. I check by multiplying back with digit sums. 846264 → 30 → 3, and 74 → 11 → 2. So 2 × (digit sum of the answer) must reduce to 3. (a) 11446 → 7, and 2 × 7 = 14 → 5. (b) 11426 → 5, and 2 × 5 = 10 → 1. (c) 11436 → 6, and 2 × 6 = 12 → 3. (d) 11336 → 5, which gives 1 again. Only (c) 11,436 works.