Here’s another question:<\/p>\n
What is the smallest positive integer y such that 71,400 multiplied by y is the square of a positive integer?<\/strong><\/p>\n * * *<\/p>\n We could spend hours doing this by trial and error. But let’s consider what we know about square numbers, and about their factors, specifically\u00a0their prime factors.<\/p>\n Example 1<\/strong> 36 = 2 x 3 x 2 x 3, or 2\u00b2 x 3\u00b2<\/p>\n Example 2<\/strong> 24 =\u00a02 x 2 x 2 x 3 so 24\u00b2 = 2 x 2 x 2 x 3 x 2 x 2 x 2 x 3 or 2^6 x 3\u00b2<\/p>\n N.B. 2^6 means 2 to the power of 6. A power is sometimes called an index or an exponent. All refer to the little number above and to the right of an integer.<\/em><\/p>\n What you will notice is that when you break a square number down into the product of its prime factors, each of those prime factors will be raised to an even power. This makes sense: even numbers are divisible by 2, and each of those powers will have to be divided by 2 to find the square root.<\/p>\n Example<\/strong> And remember that when you multiply, you add powers:<\/p>\n Example<\/strong> * * *<\/p>\n How does this help with our question? Start by factorising 71,400:<\/p>\n 71,400 = 100 x 714 = 2 x 5 x 2 x 5 x 714 = 2\u00b2 x 5\u00b2 x 2 x 357 = 2^3 x 5\u00b2 x 7 x 51 = 2^3 x 5\u00b2 x 7 x 3 x 17<\/p>\n Now, 5\u00b2 contains an even power. But the other prime factors don’t. So to get a square number, we will need to multiply once by 2, and then by 7, 3 and 17.<\/p>\n Answer: 2 x 7 x 3 x 17 = 714.<\/p>\n N.B. There are different ways of factorising. Taking out powers of 10 (and then 5 if you like) is always a good way to start: just remember to break down the 10 into 2 x 5. You can also work up through the prime numbers: 2, 3, 5, 7, 11 is normally enough.<\/p>\n * \u00a0 * \u00a0 *<\/p>\n There’s a trick to see whether a number is divisible by 3. If the sum of its digits is a multiple of 3 then the number is also a multiple of 3. This sounds more complicated than it is. Take 357:<\/p>\n 3 + 5 + 7 = 15<\/p>\n 15 is a multiple of 3<\/p>\n so 357 is a multiple of 3<\/p>\n * \u00a0 * \u00a0 *<\/p>\n Practice<\/strong> Do the same with the first few cube numbers<\/p>\n What number is both a square and a cube?<\/p>\n Change 71,400 in the question above to a) 2,835 \u00a0 b) 377,300 \u00a0 c) 168,168; \u00a0 what is y in each case?<\/p>\n","protected":false},"excerpt":{"rendered":" Here’s another question: What is the smallest positive integer y such that 71,400 multiplied by y is the square of a positive integer? * * * We could spend hours doing this by trial and error. But let’s consider what we know about square numbers, and about their factors, specifically\u00a0their prime factors. Example 1 36 … <\/p>\n
\n36 = 6 x 6. Break it down into the product of its prime factors:<\/p>\n
\n576 = 24 x 24<\/p>\n
\nThe square root of 3\u00b2 x 7^18 x 13^6 is 3 x 7^9 x 13^3.<\/p>\n
\n13^3 x 13^3 = 13^(3+3) = 13^6<\/p>\n
\nWrite out and become familiar with square numbers from 2\u00b2 = 4 up to 25\u00b2 = 625<\/p>\n