
Are my steps correct?please help thank you.
Are my steps correct?please help thank you.
A more formal way to do the working is this:
(a) sqrt(180 x 784 x h) = sqrt(22 x 32 x 5 x 24 x 72 x h) = 23 x 3 x 7 sqrt(5 x h) Hence smallest h is 5. (b) Let m = 2a x 3b x 5c x 7d. Then curt(180 x 784 x m) = curt(26+a x 32+b x 51+c x 72+d) We need exponents (6+a, 2+b, 1+c, 2+d) to be multiples of 3. The smallest values of (a,b,c,d) is (0,1,2,1) which gives exponents (6,3,3,3). Hence smallest m = 3 x 52 x 7 = 525. (c) Let k = 2a x 3b x 5c x 7d. Then curt(180 x 784 x 3 / k) = curt(26-a x 33-b x 51-c x 72-d) We need exponents (6-a, 3-b, 1-c, 2-d) to all be multiples of 3. The smallest values of (a,b,c,d) is (0,0,1,2) which gives exponents (6,3,0,0). Hence smallest k = 5 x 72 = 245.
Assuming h, m and k are positive integers (it wasn’t indicated), then all your answers are correct. Well done!
Without that assumption, h and m could be zero. 😛Â
A more formal way to do the working is this:
(a) sqrt(180 x 784 x h) = sqrt(22 x 32 x 5 x 24 x 72 x h) = 23 x 3 x 7 sqrt(5 x h) Hence smallest h is 5. (b) Let m = 2a x 3b x 5c x 7d. Then curt(180 x 784 x m) = curt(26+a x 32+b x 51+c x 72+d) We need exponents (6+a, 2+b, 1+c, 2+d) to be multiples of 3. The smallest values of (a,b,c,d) is (0,1,2,1) which gives exponents (6,3,3,3). Hence smallest m = 3 x 52 x 7 = 525. (c) Let k = 2a x 3b x 5c x 7d. Then curt(180 x 784 x 3 / k) = curt(26-a x 33-b x 51-c x 72-d) We need exponents (6-a, 3-b, 1-c, 2-d) to all be multiples of 3. The smallest values of (a,b,c,d) is (0,0,1,2) which gives exponents (6,3,0,0). Hence smallest k = 5 x 72 = 245.